Theorem necon1bbii 2569

Description: Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.)

Hypothesis

Ref	Expression
necon1bbii.1	⊢ (A ≠ B ↔ φ)

Assertion

Ref	Expression
necon1bbii	⊢ (¬ φ ↔ A = B)

Proof of Theorem necon1bbii

Step	Hyp	Ref	Expression
1		df-ne 2519	. . 3 ⊢ (A ≠ B ↔ ¬ A = B)
2		necon1bbii.1	. . 3 ⊢ (A ≠ B ↔ φ)
3	1, 2	bitr3i 242	. 2 ⊢ (¬ A = B ↔ φ)
4	3	con1bii 321	1 ⊢ (¬ φ ↔ A = B)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 176   = wceq 1642   ≠ wne 2517

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

This proof depends on definitions:  df-bi 177  df-ne 2519

This theorem is used by:  necon2bbii  2573  rabeq0  3573