Theorem aecoms-o 40052

Description: A commutation rule for identical variable specifiers. Version of aecoms 2457 using ax-c11 40037. (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 40051	. 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)

This proof depends on syntax axioms:   → wi 4  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-c5 40033  ax-c4 40034  ax-c7 40035  ax-c10 40036  ax-c11 40037  ax-c9 40040

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812

This theorem is used by:  hbae-o  40053  dral1-o  40054  dvelimf-o  40079  aev-o  40081  ax12indalem  40095  ax12inda2ALT  40096