Theorem necon3bi 2255

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 562	. 2 ⊢ (¬ 𝜑 → ¬ 𝐴 = 𝐵)
3		df-ne 2206	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
4	2, 3	sylibr 137	1 ⊢ (¬ 𝜑 → 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1243   ≠ wne 2204

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545

This theorem depends on definitions:  df-bi 110  df-ne 2206

This theorem is referenced by:  pwne  3913  nltpnft  8730  ngtmnft  8731