Theorem hbnaes 2435

Description: Rule that applies hbnae 2432 to antecedent. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 15-May-1993.) (New usage is discouraged.)

Hypothesis

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

Assertion

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

Proof of Theorem hbnaes

Step	Hyp	Ref	Expression
1		hbnae 2432	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
2		hbnaes.1	. 2 ⊢ (∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦 → 𝜑)
3	1, 2	syl 17	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787

This theorem is referenced by: (None)