Theorem pm2.21ddne 2907

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 2828	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
4	1, 3	pm2.21dd 186	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   = wceq 1523   ≠ wne 2823

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 197  df-ne 2824

This theorem is referenced by:  cshwshashlem2  15850  dprdsn  18481  coseq00topi  24299  tglndim0  25569  ncolncol  25586  footne  25660  sgnsub  30734  sgnmulsgn  30739  sgnmulsgp  30740  pconnconn  31339  osumcllem11N  35570  dochexmidlem8  37073  fnchoice  39502