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Theorem necon4bid 2582

Description: Contrapositive law deduction for inequality. (Contributed by NM, 29-Jun-2007.)

**Hypothesis**
Ref	Expression
necon4bid.1	⊢ (φ → (A ≠ B ↔ C ≠ D))

**Assertion**
Ref	Expression
necon4bid	⊢ (φ → (A = B ↔ C = D))

**Proof of Theorem necon4bid**
Step	Hyp	Ref	Expression
1		necon4bid.1	. . 3 ⊢ (φ → (A ≠ B ↔ C ≠ D))
2	1	necon2bbid 2574	. 2 ⊢ (φ → (C = D ↔ ¬ A ≠ B))
3		nne 2520	. 2 ⊢ (¬ A ≠ B ↔ A = B)
4	2, 3	syl6rbb 253	1 ⊢ (φ → (A = B ↔ C = D))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ 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: nebi 2587