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Theorem aecom-o 34690
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 2453 using ax-c11 34676. Unlike axc11nfromc11 34715, this version does not require ax-5 1988 (see comment of equcomi1 34689). (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 34676 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
21pm2.43i 52 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
3 equcomi1 34689 . . 3 (𝑥 = 𝑦𝑦 = 𝑥)
43alimi 1888 . 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 1630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-c5 34672  ax-c4 34673  ax-c7 34674  ax-c10 34675  ax-c11 34676  ax-c9 34679
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854
This theorem is referenced by:  aecoms-o  34691  naecoms-o  34716  aev-o  34720  ax12indalem  34734
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