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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 = 𝑦𝜑)
Assertion
Ref Expression
hbnaes (¬ ∀𝑥 𝑥 = 𝑦𝜑)

Proof of Theorem hbnaes
StepHypRef Expression
1 hbnae 1609 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
2 hbnalequs.1 . 2 (∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦𝜑)
31, 2syl 14 1 (¬ ∀𝑥 𝑥 = 𝑦𝜑)
 Colors of variables: wff set class 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
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