Theorem necon2ad 2384

Description: Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)

Hypothesis

Ref	Expression
necon2ad.1	⊢ (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓))

Assertion

Ref	Expression
necon2ad	⊢ (𝜑 → (𝜓 → 𝐴 ≠ 𝐵))

Proof of Theorem necon2ad

Step	Hyp	Ref	Expression
1		necon2ad.1	. . 3 ⊢ (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓))
2	1	con2d 614	. 2 ⊢ (𝜑 → (𝜓 → ¬ 𝐴 = 𝐵))
3		df-ne 2328	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
4	2, 3	syl6ibr 161	1 ⊢ (𝜑 → (𝜓 → 𝐴 ≠ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1335   ≠ wne 2327

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 2328

This theorem is referenced by:  necon2d  2386  prneimg  3737  tz7.2  4314  nordeq  4502  pr2ne  7122  ltne  7957  apne  8493  xrltne  9712  npnflt  9714  nmnfgt  9717  ge0nemnf  9723  rpexp  12022  sqrt2irr  12031