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Theorem axc11nfromc11 33229
 Description: Rederivation of ax-c11n 33191 from original version ax-c11 33190. See theorem axc11 2302 for the derivation of ax-c11 33190 from ax-c11n 33191. This theorem should not be referenced in any proof. Instead, use ax-c11n 33191 above so that uses of ax-c11n 33191 can be more easily identified, or use aecom-o 33204 when this form is needed for studies involving ax-c11 33190 and omitting ax-5 1827. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axc11nfromc11 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Proof of Theorem axc11nfromc11
StepHypRef Expression
1 ax-c11 33190 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
21pm2.43i 50 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
3 equcomi 1931 . . 3 (𝑥 = 𝑦𝑦 = 𝑥)
43alimi 1730 . 2 (∀𝑦 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
52, 4syl 17 1 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-c11 33190 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696 This theorem is referenced by: (None)
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