Theorem aecoms-o 39644

Description: A commutation rule for identical variable specifiers. Version of aecoms 2458 using ax-c11 39629. (Contributed by NM, 10-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
alequcoms-o.1	⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑)

Assertion

Ref	Expression
aecoms-o	⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑)

Proof of Theorem aecoms-o

Step	Hyp	Ref	Expression
1		aecom-o 39643	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → ∀𝑥 𝑥 = 𝑦)
2		alequcoms-o.1	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑)
3	1, 2	syl 18	1 ⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑)

Syntax hints:   → wi 4  ∀wal 1566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-c5 39625  ax-c4 39626  ax-c7 39627  ax-c10 39628  ax-c11 39629  ax-c9 39632

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1808

This theorem is referenced by:  hbae-o  39645  dral1-o  39646  dvelimf-o  39671  aev-o  39673  ax12indalem  39687  ax12inda2ALT  39688