Theorem necon3ad 2399

Description: Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)

Hypothesis

Ref	Expression
necon3ad.1	⊢ (𝜑 → (𝜓 → 𝐴 = 𝐵))

Assertion

Ref	Expression
necon3ad	⊢ (𝜑 → (𝐴 ≠ 𝐵 → ¬ 𝜓))

Proof of Theorem necon3ad

Step	Hyp	Ref	Expression
1		df-ne 2358	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		necon3ad.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝐴 = 𝐵))
3	2	con3d 632	. 2 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 → ¬ 𝜓))
4	1, 3	biimtrid 152	1 ⊢ (𝜑 → (𝐴 ≠ 𝐵 → ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1363   ≠ wne 2357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-in1 615  ax-in2 616

This theorem depends on definitions:  df-bi 117  df-ne 2358

This theorem is referenced by:  necon3d  2401  disjnim  4006  fodjumkvlemres  7171  nlt1pig  7354  eucalglt  12071  nprm  12137  pcprmpw2  12346  pcmpt  12355  expnprm  12365  0nnei  13949  2sqlem8a  14765  bj-charfunr  14858