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Theorem axc11-o 38124
Description: Show that ax-c11 38060 can be derived from ax-c11n 38061 and ax-12 2169. An open problem is whether this theorem can be derived from ax-c11n 38061 and the others when ax-12 2169 is replaced with ax-c15 38062 or ax12v 2170. See Theorems axc11nfromc11 38099 for the rederivation of ax-c11n 38061 from axc11 2427.

Normally, axc11 2427 should be used rather than ax-c11 38060 or axc11-o 38124, 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 38061 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
2 ax12 2420 . . . 4 (𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥𝜑)))
32equcoms 2021 . . 3 (𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥𝜑)))
43sps-o 38081 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥𝜑)))
5 pm2.27 42 . . 3 (𝑦 = 𝑥 → ((𝑦 = 𝑥𝜑) → 𝜑))
65al2imi 1815 . 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 1537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-10 2135  ax-12 2169  ax-13 2369  ax-c5 38056  ax-c11n 38061
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1780  df-nf 1784
This theorem is referenced by: (None)
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