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

Theorem axc11nfromc11 38335
Description: Rederivation of ax-c11n 38297 from original version ax-c11 38296. See Theorem axc11 2424 for the derivation of ax-c11 38296 from ax-c11n 38297.

This theorem should not be referenced in any proof. Instead, use ax-c11n 38297 above so that uses of ax-c11n 38297 can be more easily identified, or use aecom-o 38310 when this form is needed for studies involving ax-c11 38296 and omitting ax-5 1906. (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 38296 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
21pm2.43i 52 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
3 equcomi 2013 . . 3 (𝑥 = 𝑦𝑦 = 𝑥)
43alimi 1806 . 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 1532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-c11 38296
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator