Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  axc11nfromc11 Structured version   Visualization version   GIF version

Theorem axc11nfromc11 36536
 Description: Rederivation of ax-c11n 36498 from original version ax-c11 36497. See Theorem axc11 2441 for the derivation of ax-c11 36497 from ax-c11n 36498. This theorem should not be referenced in any proof. Instead, use ax-c11n 36498 above so that uses of ax-c11n 36498 can be more easily identified, or use aecom-o 36511 when this form is needed for studies involving ax-c11 36497 and omitting ax-5 1911. (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 36497 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
21pm2.43i 52 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
3 equcomi 2024 . . 3 (𝑥 = 𝑦𝑦 = 𝑥)
43alimi 1813 . 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 1536 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-c11 36497 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator