Theorem naecoms-o 39556

Description: A commutation rule for distinct variable specifiers. Version of naecoms 2461 using ax-c11 39516. (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 39530	. . 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 1559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-c5 39512  ax-c4 39513  ax-c7 39514  ax-c10 39515  ax-c11 39516  ax-c9 39519

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1801

This theorem is referenced by:  ax12inda2ALT  39575