Theorem necon3bd 2420

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

Hypothesis

Ref	Expression
necon3bd.1	⊢ (𝜑 → (𝐴 = 𝐵 → 𝜓))

Assertion

Ref	Expression
necon3bd	⊢ (𝜑 → (¬ 𝜓 → 𝐴 ≠ 𝐵))

Proof of Theorem necon3bd

Step	Hyp	Ref	Expression
1		necon3bd.1	. . 3 ⊢ (𝜑 → (𝐴 = 𝐵 → 𝜓))
2	1	con3d 632	. 2 ⊢ (𝜑 → (¬ 𝜓 → ¬ 𝐴 = 𝐵))
3		df-ne 2378	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
4	2, 3	imbitrrdi 162	1 ⊢ (𝜑 → (¬ 𝜓 → 𝐴 ≠ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1373   ≠ wne 2377

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 615  ax-in2 616

This theorem depends on definitions:  df-bi 117  df-ne 2378

This theorem is referenced by:  nelne1  2467  nelne2  2468  nssne1  3253  nssne2  3254  disjne  3516  difsn  3773  nbrne1  4067  nbrne2  4068  ac6sfi  7007  indpi  7468  zneo  9487  pc2dvds  12703  pcadd  12713  oddprmdvds  12727  4sqlem11  12774  isnzr2  13996  lssvneln0  14185  lgsne0  15565  lgsquadlem2  15605  lgsquadlem3  15606