Theorem necon3bd 2443

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 634	. 2 ⊢ (𝜑 → (¬ 𝜓 → ¬ 𝐴 = 𝐵))
3		df-ne 2401	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
4	2, 3	imbitrrdi 162	1 ⊢ (𝜑 → (¬ 𝜓 → 𝐴 ≠ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1395   ≠ wne 2400

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 617  ax-in2 618

This theorem depends on definitions:  df-bi 117  df-ne 2401

This theorem is referenced by:  nelne1  2490  nelne2  2491  nssne1  3283  nssne2  3284  disjne  3546  difsn  3806  nbrne1  4103  nbrne2  4104  ac6sfi  7078  indpi  7550  zneo  9569  pc2dvds  12890  pcadd  12900  oddprmdvds  12914  4sqlem11  12961  isnzr2  14185  lssvneln0  14374  lgsne0  15754  lgsquadlem2  15794  lgsquadlem3  15795