Theorem equs3 2006

Description: Lemma used in proofs of substitution properties. (Contributed by NM, 10-May-1993.)

Assertion

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

Proof of Theorem equs3

Step	Hyp	Ref	Expression
1		alinexa 1888	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → ¬ 𝜑) ↔ ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
2	1	con2bii 349	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ¬ ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386  ∀wal 1599  ∃wex 1823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824

This theorem is referenced by: (None)