Theorem necon3ad 2409

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 2368	. 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 1364   ≠ wne 2367

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 2368

This theorem is referenced by:  necon3d  2411  disjnim  4024  fodjumkvlemres  7225  nlt1pig  7408  eucalglt  12225  nprm  12291  pcprmpw2  12502  pcmpt  12512  expnprm  12522  0nnei  14389  2sqlem8a  15363  bj-charfunr  15456