Theorem pm2.21ddne 2503

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 2441	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
4	1, 3	pm2.21dd 629	1 ⊢ (𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4   = wceq 1402   ≠ wne 2420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-in2 624

This proof depends on definitions:  df-bi 117  df-ne 2421

This theorem is used by:  npnflt  10219  nmnfgt  10222  xlt2add  10284  xrbdtri  12044  divalglemex  12691  divalg  12693  znege1  12958  ennnfonelemex  13307  pw1ndom3  17032