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Theorem necon2ai 2561

Description: Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

**Hypothesis**
Ref	Expression
necon2ai.1	⊢ (A = B → ¬ φ)

**Assertion**
Ref	Expression
necon2ai	⊢ (φ → A ≠ B)

**Proof of Theorem necon2ai**
Step	Hyp	Ref	Expression
1		nne 2520	. . 3 ⊢ (¬ A ≠ B ↔ A = B)
2		necon2ai.1	. . 3 ⊢ (A = B → ¬ φ)
3	1, 2	sylbi 187	. 2 ⊢ (¬ A ≠ B → ¬ φ)
4	3	con4i 122	1 ⊢ (φ → 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: necon2i 2563 neneqad 2586 peano4 4557 addccan2 4559 ncssfin 6151 ncspw1eu 6159 nntccl 6170 ceclb 6183 ce0ncpw1 6185 cecl 6186 nclecid 6197 le0nc 6200 addlecncs 6209