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Theorem aecom 2299
 Description: Commutation law for identical variable specifiers. Both sides of the biconditional are true when 𝑥 and 𝑦 are substituted with the same variable. (Contributed by NM, 10-May-1993.) Changed to a biconditional. (Revised by BJ, 26-Sep-2019.)
Assertion
Ref Expression
aecom (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑦 𝑦 = 𝑥)

Proof of Theorem aecom
StepHypRef Expression
1 axc11n 2295 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
2 axc11n 2295 . 2 (∀𝑦 𝑦 = 𝑥 → ∀𝑥 𝑥 = 𝑦)
31, 2impbii 198 1 (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑦 𝑦 = 𝑥)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195  ∀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-10 2006  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701 This theorem is referenced by:  aecoms  2300  naecoms  2301  wl-nfae1  32494
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