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Theorem necon2bbii 2572

Description: Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)

**Hypothesis**
Ref	Expression
necon2bbii.1	⊢ (φ ↔ A ≠ B)

**Assertion**
Ref	Expression
necon2bbii	⊢ (A = B ↔ ¬ φ)

**Proof of Theorem necon2bbii**
Step	Hyp	Ref	Expression
1		necon2bbii.1	. . . 4 ⊢ (φ ↔ A ≠ B)
2	1	bicomi 193	. . 3 ⊢ (A ≠ B ↔ φ)
3	2	necon1bbii 2568	. 2 ⊢ (¬ φ ↔ A = B)
4	3	bicomi 193	1 ⊢ (A = B ↔ ¬ φ)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 176 = wceq 1642 ≠ wne 2516

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

This theorem depends on definitions: df-bi 177 df-ne 2518

This theorem is referenced by: dfaddc2 4381 xpeq0 5046