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

Theorem aecom-o 35917
Description: Commutation law for identical variable specifiers. The antecedent and consequent are true when 𝑥 and 𝑦 are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). Version of aecom 2441 using ax-c11 35903. Unlike axc11nfromc11 35942, this version does not require ax-5 1902 (see comment of equcomi1 35916). (Contributed by NM, 10-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
aecom-o (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Proof of Theorem aecom-o
StepHypRef Expression
1 ax-c11 35903 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
21pm2.43i 52 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
3 equcomi1 35916 . . 3 (𝑥 = 𝑦𝑦 = 𝑥)
43alimi 1803 . 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 1526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-c5 35899  ax-c4 35900  ax-c7 35901  ax-c10 35902  ax-c11 35903  ax-c9 35906
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772
This theorem is referenced by:  aecoms-o  35918  naecoms-o  35943  aev-o  35947  ax12indalem  35961
  Copyright terms: Public domain W3C validator