Theorem naecoms-o 39729

Description: A commutation rule for distinct variable specifiers. Version of naecoms 2460 using ax-c11 39689. (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 39703	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
2		nalequcoms-o.1	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → 𝜑)
3	1, 2	nsyl4 159	. 2 ⊢ (¬ 𝜑 → ∀𝑦 𝑦 = 𝑥)
4	3	con1i 148	1 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-c5 39685  ax-c4 39686  ax-c7 39687  ax-c10 39688  ax-c11 39689  ax-c9 39692

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809

This theorem is used by:  ax12inda2ALT  39748