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Theorem necon3bd 2553

Description: Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

**Hypothesis**
Ref	Expression
necon3bd.1	⊢ (φ → (A = B → ψ))

**Assertion**
Ref	Expression
necon3bd	⊢ (φ → (¬ ψ → A ≠ B))

**Proof of Theorem necon3bd**
Step	Hyp	Ref	Expression
1		nne 2520	. . 3 ⊢ (¬ A ≠ B ↔ A = B)
2		necon3bd.1	. . 3 ⊢ (φ → (A = B → ψ))
3	1, 2	syl5bi 208	. 2 ⊢ (φ → (¬ A ≠ B → ψ))
4	3	con1d 116	1 ⊢ (φ → (¬ ψ → A ≠ B))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 = 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: nelne1 2605 nelne2 2606 nssne1 3327 nssne2 3328 disjne 3596 difsn 3845 nbrne1 4656 nbrne2 4657