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Theorem aecom-o 34505
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 2344 using ax-c11 34491. Unlike axc11nfromc11 34530, this version does not require ax-5 1879 (see comment of equcomi1 34504). (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 34491 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
21pm2.43i 52 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
3 equcomi1 34504 . . 3 (𝑥 = 𝑦𝑦 = 𝑥)
43alimi 1779 . 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 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-c5 34487  ax-c4 34488  ax-c7 34489  ax-c10 34490  ax-c11 34491  ax-c9 34494
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745
This theorem is referenced by:  aecoms-o  34506  naecoms-o  34531  aev-o  34535  ax12indalem  34549
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