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Theorem aecom-o 37413
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 2426 using ax-c11 37399. Unlike axc11nfromc11 37438, this version does not require ax-5 1914 (see comment of equcomi1 37412). (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 37399 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
21pm2.43i 52 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
3 equcomi1 37412 . . 3 (𝑥 = 𝑦𝑦 = 𝑥)
43alimi 1814 . 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 1540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-c5 37395  ax-c4 37396  ax-c7 37397  ax-c10 37398  ax-c11 37399  ax-c9 37402
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783
This theorem is referenced by:  aecoms-o  37414  naecoms-o  37439  aev-o  37443  ax12indalem  37457
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