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Theorem aecom 2406
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.) Change 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 2405 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
2 axc11n 2405 . 2 (∀𝑦 𝑦 = 𝑥 → ∀𝑥 𝑥 = 𝑦)
31, 2impbii 210 1 (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑦 𝑦 = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wal 1520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-10 2112  ax-12 2141  ax-13 2344
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1762  df-nf 1766
This theorem is referenced by:  aecoms  2407  naecoms  2408  wl-nfae1  34326
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