Theorem equs3 1644

Description: Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
equs3	⊢ (∃x(x = y ∧ φ) ↔ ¬ ∀x(x = y → ¬ φ))

Proof of Theorem equs3

Step	Hyp	Ref	Expression
1		alinexa 1578	. 2 ⊢ (∀x(x = y → ¬ φ) ↔ ¬ ∃x(x = y ∧ φ))
2	1	con2bii 322	1 ⊢ (∃x(x = y ∧ φ) ↔ ¬ ∀x(x = y → ¬ φ))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is used by:  equs5e  1888  sbn  2062