Theorem pm2.21ddne 2419

Description: A contradiction implies anything. Equality/inequality deduction form. (Contributed by David Moews, 28-Feb-2017.)

Hypotheses

Ref	Expression
pm2.21ddne.1	⊢ (𝜑 → 𝐴 = 𝐵)
pm2.21ddne.2	⊢ (𝜑 → 𝐴 ≠ 𝐵)

Assertion

Ref	Expression
pm2.21ddne	⊢ (𝜑 → 𝜓)

Proof of Theorem pm2.21ddne

Step	Hyp	Ref	Expression
1		pm2.21ddne.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		pm2.21ddne.2	. . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
3	2	neneqd 2357	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
4	1, 3	pm2.21dd 610	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   = wceq 1343   ≠ wne 2336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-in2 605

This theorem depends on definitions:  df-bi 116  df-ne 2337

This theorem is referenced by:  npnflt  9751  nmnfgt  9754  xlt2add  9816  xrbdtri  11217  divalglemex  11859  divalg  11861  znege1  12110  ennnfonelemex  12347