Theorem hbnaes 1716

Description: Rule that applies hbnae 1714 to antecedent. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
hbnalequs.1	⊢ (∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦 → 𝜑)

Assertion

Ref	Expression
hbnaes	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → 𝜑)

Proof of Theorem hbnaes

Step	Hyp	Ref	Expression
1		hbnae 1714	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
2		hbnalequs.1	. 2 ⊢ (∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦 → 𝜑)
3	1, 2	syl 14	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354

This theorem is referenced by:  sbal2  2013