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Theorem hbnaes 2443
Description: Rule that applies hbnae 2440 to antecedent. Usage of this theorem is discouraged because it depends on ax-13 2380. (Contributed by NM, 15-May-1993.) (New usage is discouraged.)
Hypothesis
Ref Expression
hbnaes.1 (∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦𝜑)
Assertion
Ref Expression
hbnaes (¬ ∀𝑥 𝑥 = 𝑦𝜑)

Proof of Theorem hbnaes
StepHypRef Expression
1 hbnae 2440 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
2 hbnaes.1 . 2 (∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦𝜑)
31, 2syl 17 1 (¬ ∀𝑥 𝑥 = 𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2158  ax-12 2178  ax-13 2380
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782
This theorem is referenced by: (None)
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