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Theorem axc11-o 34555
 Description: Show that ax-c11 34491 can be derived from ax-c11n 34492 and ax-12 2087. An open problem is whether this theorem can be derived from ax-c11n 34492 and the others when ax-12 2087 is replaced with ax-c15 34493 or ax12v 2088. See theorem axc11nfromc11 34530 for the rederivation of ax-c11n 34492 from axc11 2347. Normally, axc11 2347 should be used rather than ax-c11 34491 or axc11-o 34555, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc11-o (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Proof of Theorem axc11-o
StepHypRef Expression
1 ax-c11n 34492 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
2 ax12 2340 . . . 4 (𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥𝜑)))
32equcoms 1993 . . 3 (𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥𝜑)))
43sps-o 34512 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥𝜑)))
5 pm2.27 42 . . 3 (𝑦 = 𝑥 → ((𝑦 = 𝑥𝜑) → 𝜑))
65al2imi 1783 . 2 (∀𝑦 𝑦 = 𝑥 → (∀𝑦(𝑦 = 𝑥𝜑) → ∀𝑦𝜑))
71, 4, 6sylsyld 61 1 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1521 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087  ax-13 2282  ax-c5 34487  ax-c11n 34492 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750 This theorem is referenced by: (None)
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