In exodus 7 it says on verses 8-12:


  ח וַיֹּאמֶר יְהוָה, אֶל-מֹשֶׁה וְאֶל-אַהֲרֹן לֵאמֹר.  ט כִּי יְדַבֵּר אֲלֵכֶם פַּרְעֹה לֵאמֹר, תְּנוּ לָכֶם מוֹפֵת; וְאָמַרְתָּ אֶל-אַהֲרֹן, קַח אֶת-מַטְּךָ וְהַשְׁלֵךְ לִפְנֵי-פַרְעֹה--יְהִי לְתַנִּין.  י וַיָּבֹא מֹשֶׁה וְאַהֲרֹן, אֶל-פַּרְעֹה, וַיַּעֲשׂוּ כֵן, כַּאֲשֶׁר צִוָּה יְהוָה; וַיַּשְׁלֵךְ אַהֲרֹן אֶת-מַטֵּהוּ, לִפְנֵי פַרְעֹה וְלִפְנֵי עֲבָדָיו--וַיְהִי לְתַנִּין.  יא וַיִּקְרָא, גַּם-פַּרְעֹה, לַחֲכָמִים, וְלַמְכַשְּׁפִים; וַיַּעֲשׂוּ גַם-הֵם חַרְטֻמֵּי מִצְרַיִם, בְּלַהֲטֵיהֶם--כֵּן.  יב וַיַּשְׁלִיכוּ אִישׁ מַטֵּהוּ, וַיִּהְיוּ לְתַנִּינִם; וַיִּבְלַע מַטֵּה-אַהֲרֹן, אֶת-מַטֹּתָם


Clearly talking about the staffs of both Aaron and the Egyptian sorcerers turning to crocodiles. Yet, later on verse 15 it says:


  לֵךְ אֶל-פַּרְעֹה בַּבֹּקֶר, הִנֵּה יֹצֵא הַמַּיְמָה, וְנִצַּבְתָּ לִקְרָאתוֹ, עַל-שְׂפַת הַיְאֹר; וְהַמַּטֶּה אֲשֶׁר-נֶהְפַּךְ לְנָחָשׁ, תִּקַּח בְּיָדֶךָ.


Talking about the staff that turned into a snake. Never before any staff turned into a snake. And the only staff that turned into something before was Aaron's staff, that turned into a crocodile.

What exactly did Aaron's staff turned into? And what is the reason for the difference in the animal?


Question: Is this formula in predicate logic a tautology?
Answer: It is valid; after restoring the parentheses for better readibility, we have :


  $(∀x[p(x) \to q(x)] ∧ p(y)) \to q(y)$.


Proof

1) $∀x[p(x) \to q(x)] ∧ p(y)$ --- assumed

2) $∀x[p(x) \to q(x)]$ --- from 1) by b$\land$-elimination

3) $p(y)$ --- from 1) by b$\land$-elimination

4) $p(y) \to q(y)$ --- from 2) by $\forall$-elimination

5) $q(y)$ --- from 3) and 4) by $\to$-elimination


  6) $(∀x[p(x) \to q(x)] ∧ p(y)) \to q(y)$ --- from 1) and 5) by $\to$-introduction.