Theorem necon2ad 2437

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 627	. 2 ⊢ (𝜑 → (𝜓 → ¬ 𝐴 = 𝐵))
3		df-ne 2381	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
4	2, 3	imbitrrdi 162	1 ⊢ (𝜑 → (𝜓 → 𝐴 ≠ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1375   ≠ wne 2380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618

This theorem depends on definitions:  df-bi 117  df-ne 2381

This theorem is referenced by:  necon2d  2439  prneimg  3831  tz7.2  4422  nordeq  4613  pr2ne  7333  ltne  8199  apne  8738  xrltne  9977  npnflt  9979  nmnfgt  9982  ge0nemnf  9988  rpexp  12641  sqrt2irr  12650  pcgcd1  12817  nzrunit  14117  lgsmod  15670