Theorem naecoms-o 2178

Description: A commutation rule for distinct variable specifiers. Version of naecoms 1948 using ax-10o 2139. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

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

Assertion

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

Proof of Theorem naecoms-o

Step	Hyp	Ref	Expression
1		aecom-o 2151	. . 3 ⊢ (∀x x = y → ∀y y = x)
2		nalequcoms-o.1	. . 3 ⊢ (¬ ∀x x = y → φ)
3	1, 2	nsyl4 134	. 2 ⊢ (¬ φ → ∀y y = x)
4	3	con1i 121	1 ⊢ (¬ ∀y y = x → φ)

Syntax hints:  ¬ wn 3   → 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:  ax11inda2ALT  2198