Theorem pm2.21ddne 3055

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 2976	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
4	1, 3	pm2.21dd 187	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   = wceq 1653   ≠ wne 2971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 199  df-ne 2972

This theorem is referenced by:  cshwshashlem2  16131  dprdsn  18751  coseq00topi  24596  tglndim0  25880  ncolncol  25897  footne  25971  sgnsub  31123  sgnmulsgn  31128  sgnmulsgp  31129  pconnconn  31730  osumcllem11N  35987  dochexmidlem8  37488  fnchoice  39948