Theorem hbnaes 1957

Description: Rule that applies hbnae 1955 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 1955	. 2 ⊢ (¬ ∀x x = y → ∀z ¬ ∀x x = y)
2		hbnalequs.1	. 2 ⊢ (∀z ¬ ∀x x = y → φ)
3	1, 2	syl 15	1 ⊢ (¬ ∀x x = y → φ)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  sbal1  2126