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Theorem necon1ai 2558

Description: Contrapositive inference for inequality. (Contributed by NM, 12-Feb-2007.)

**Hypothesis**
Ref	Expression
necon1ai.1	⊢ (¬ φ → A = B)

**Assertion**
Ref	Expression
necon1ai	⊢ (A ≠ B → φ)

**Proof of Theorem necon1ai**
Step	Hyp	Ref	Expression
1		df-ne 2518	. 2 ⊢ (A ≠ B ↔ ¬ A = B)
2		necon1ai.1	. . 3 ⊢ (¬ φ → A = B)
3	2	con1i 121	. 2 ⊢ (¬ A = B → φ)
4	1, 3	sylbi 187	1 ⊢ (A ≠ B → φ)

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: necon1i 2560 tz6.12i 5348 elfvdm 5351 elovex12 5648