Theorem aecoms-o 2152

Description: A commutation rule for identical variable specifiers. Version of aecoms 1947 using ax-10o . (Contributed by NM, 5-Aug-1993.) (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 2151	. 2 ⊢ (∀y y = x → ∀x x = y)
2		alequcoms-o.1	. 2 ⊢ (∀x x = y → φ)
3	1, 2	syl 15	1 ⊢ (∀y y = x → φ)

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 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-10o 2139

This theorem depends on definitions:  df-bi 177  df-ex 1542

This theorem is referenced by:  hbae-o  2153  dral1-o  2154  dvelimf-o  2180  aev-o  2182  ax11indalem  2197  ax11inda2ALT  2198