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Theorem aecom-o 34976
 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 2448 using ax-c11 34962. Unlike axc11nfromc11 35001, this version does not require ax-5 2011 (see comment of equcomi1 34975). (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 34962 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
21pm2.43i 52 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
3 equcomi1 34975 . . 3 (𝑥 = 𝑦𝑦 = 𝑥)
43alimi 1912 . 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 1656 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-c5 34958  ax-c4 34959  ax-c7 34960  ax-c10 34961  ax-c11 34962  ax-c9 34965 This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1881 This theorem is referenced by:  aecoms-o  34977  naecoms-o  35002  aev-o  35006  ax12indalem  35020
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