Theorem necon3bi 2386

Description: Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)

Hypothesis

Ref	Expression
necon3bi.1	⊢ (𝐴 = 𝐵 → 𝜑)

Assertion

Ref	Expression
necon3bi	⊢ (¬ 𝜑 → 𝐴 ≠ 𝐵)

Proof of Theorem necon3bi

Step	Hyp	Ref	Expression
1		necon3bi.1	. . 3 ⊢ (𝐴 = 𝐵 → 𝜑)
2	1	con3i 622	. 2 ⊢ (¬ 𝜑 → ¬ 𝐴 = 𝐵)
3		df-ne 2337	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
4	2, 3	sylibr 133	1 ⊢ (¬ 𝜑 → 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1343   ≠ wne 2336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605

This theorem depends on definitions:  df-bi 116  df-ne 2337

This theorem is referenced by:  pwne  4139  sucpw1ne3  7188  nltpnft  9750  ngtmnft  9753