Theorem necon3bd 2457

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 636	. 2 ⊢ (𝜑 → (¬ 𝜓 → ¬ 𝐴 = 𝐵))
3		df-ne 2415	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
4	2, 3	imbitrrdi 162	1 ⊢ (𝜑 → (¬ 𝜓 → 𝐴 ≠ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1398   ≠ wne 2414

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 619  ax-in2 620

This theorem depends on definitions:  df-bi 117  df-ne 2415

This theorem is referenced by:  nelne1  2504  nelne2  2505  nssne1  3300  nssne2  3301  disjne  3566  difsn  3836  nbrne1  4133  nbrne2  4134  ac6sfi  7168  indpi  7673  zneo  9697  pc2dvds  13053  pcadd  13063  oddprmdvds  13077  4sqlem11  13124  isnzr2  14414  lssvneln0  14633  pellexlem1  15957  lgsne0  16023  lgsquadlem2  16063  lgsquadlem3  16064