Theorem necon3ad 2350

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 2309	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		necon3ad.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝐴 = 𝐵))
3	2	con3d 620	. 2 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 → ¬ 𝜓))
4	1, 3	syl5bi 151	1 ⊢ (𝜑 → (𝐴 ≠ 𝐵 → ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1331   ≠ wne 2308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-in1 603  ax-in2 604

This theorem depends on definitions:  df-bi 116  df-ne 2309

This theorem is referenced by:  necon3d  2352  disjnim  3920  fodjumkvlemres  7033  nlt1pig  7149  eucalglt  11738  nprm  11804  0nnei  12322