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Theorem axc5c711to11 33048
Description: Rederivation of ax-11 2020 from axc5c711 33045. Note that ax-c7 33012 and ax-11 2020 are not used by the rederivation. The use of alimi 1729 (which uses ax-c5 33010) is allowed since we have already proved axc5c711toc5 33046. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc5c711to11 (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Proof of Theorem axc5c711to11
StepHypRef Expression
1 axc5c711toc7 33047 . . 3 (¬ ∀𝑦 ¬ ∀𝑦 ¬ ∀𝑥𝑦𝜑 → ¬ ∀𝑥𝑦𝜑)
21con4i 111 . 2 (∀𝑥𝑦𝜑 → ∀𝑦 ¬ ∀𝑦 ¬ ∀𝑥𝑦𝜑)
3 pm2.21 118 . . . . . 6 (¬ ∀𝑥𝑦 ¬ ∀𝑥𝑦𝜑 → (∀𝑥𝑦 ¬ ∀𝑥𝑦𝜑 → ∀𝑥𝜑))
4 axc5c711 33045 . . . . . 6 ((∀𝑥𝑦 ¬ ∀𝑥𝑦𝜑 → ∀𝑥𝜑) → 𝜑)
53, 4syl 17 . . . . 5 (¬ ∀𝑥𝑦 ¬ ∀𝑥𝑦𝜑𝜑)
65alimi 1729 . . . 4 (∀𝑥 ¬ ∀𝑥𝑦 ¬ ∀𝑥𝑦𝜑 → ∀𝑥𝜑)
7 axc5c711toc7 33047 . . . 4 (¬ ∀𝑥 ¬ ∀𝑥𝑦 ¬ ∀𝑥𝑦𝜑 → ∀𝑦 ¬ ∀𝑥𝑦𝜑)
86, 7nsyl4 154 . . 3 (¬ ∀𝑦 ¬ ∀𝑥𝑦𝜑 → ∀𝑥𝜑)
98alimi 1729 . 2 (∀𝑦 ¬ ∀𝑦 ¬ ∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
102, 9syl 17 1 (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-11 2020  ax-c5 33010  ax-c4 33011  ax-c7 33012
This theorem is referenced by: (None)
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