Theorem naecoms-o 34995

Description: A commutation rule for distinct variable specifiers. Version of naecoms 2450 using ax-c11 34955. (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 34969	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
2		nalequcoms-o.1	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → 𝜑)
3	1, 2	nsyl4 158	. 2 ⊢ (¬ 𝜑 → ∀𝑦 𝑦 = 𝑥)
4	3	con1i 147	1 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-c5 34951  ax-c4 34952  ax-c7 34953  ax-c10 34954  ax-c11 34955  ax-c9 34958

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1879

This theorem is referenced by:  ax12inda2ALT  35014