Theorem aecoms-o 40151

Description: A commutation rule for identical variable specifiers. Version of aecoms 2460 using ax-c11 40136. (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 40150	. 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 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-c5 40132  ax-c4 40133  ax-c7 40134  ax-c10 40135  ax-c11 40136  ax-c9 40139

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by:  hbae-o  40152  dral1-o  40153  dvelimf-o  40178  aev-o  40180  ax12indalem  40194  ax12inda2ALT  40195