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Theorem ax11-pm 34148
Description: Proof of ax-11 2153 similar to PM's proof of alcom 2155 (PM*11.2). For a proof closer to PM's proof, see ax11-pm2 34152. Axiom ax-11 2153 is used in the proof only through nfa2 2168. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
Assertion
Ref Expression
ax11-pm (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Proof of Theorem ax11-pm
StepHypRef Expression
1 2sp 2177 . . 3 (∀𝑥𝑦𝜑𝜑)
21gen2 1790 . 2 𝑦𝑥(∀𝑥𝑦𝜑𝜑)
3 nfa2 2168 . . 3 𝑦𝑥𝑦𝜑
4 nfa1 2148 . . 3 𝑥𝑥𝑦𝜑
53, 42stdpc5 34145 . 2 (∀𝑦𝑥(∀𝑥𝑦𝜑𝜑) → (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑))
62, 5ax-mp 5 1 (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-10 2138  ax-11 2153  ax-12 2169
This theorem depends on definitions:  df-bi 209  df-or 844  df-ex 1774  df-nf 1778
This theorem is referenced by: (None)
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