Theorem naecoms-o 36057

Description: A commutation rule for distinct variable specifiers. Version of naecoms 2447 using ax-c11 36017. (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 36031	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
2		nalequcoms-o.1	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → 𝜑)
3	1, 2	nsyl4 161	. 2 ⊢ (¬ 𝜑 → ∀𝑦 𝑦 = 𝑥)
4	3	con1i 149	1 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-c5 36013  ax-c4 36014  ax-c7 36015  ax-c10 36016  ax-c11 36017  ax-c9 36020

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777

This theorem is referenced by:  ax12inda2ALT  36076