Theorem hbnaes 1611

Description: Rule that applies hbnae 1609 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 1609	. 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 1241

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249

This theorem is referenced by:  sbal2  1898