MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  aecoms Structured version   Visualization version   GIF version

Theorem aecoms 2345
Description: A commutation rule for identical variable specifiers. (Contributed by NM, 10-May-1993.)
Hypothesis
Ref Expression
aecoms.1 (∀𝑥 𝑥 = 𝑦𝜑)
Assertion
Ref Expression
aecoms (∀𝑦 𝑦 = 𝑥𝜑)

Proof of Theorem aecoms
StepHypRef Expression
1 aecom 2344 . 2 (∀𝑦 𝑦 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
2 aecoms.1 . 2 (∀𝑥 𝑥 = 𝑦𝜑)
31, 2sylbi 207 1 (∀𝑦 𝑦 = 𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750
This theorem is referenced by:  axc11  2347  nd4  9450  axrepnd  9454  axpownd  9461  axregnd  9464  axinfnd  9466  axacndlem5  9471  axacnd  9472  wl-ax11-lem1  33492  wl-ax11-lem3  33494  wl-ax11-lem9  33500  wl-ax11-lem10  33501  e2ebind  39096
  Copyright terms: Public domain W3C validator