Theorem pm2.21ddne 2590

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 2532	. 2 ⊢ (φ → ¬ A = B)
4	1, 3	pm2.21dd 99	1 ⊢ (φ → ψ)

Syntax hints:   → wi 4   = wceq 1642   ≠ wne 2516

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 177  df-ne 2518

This theorem is referenced by: (None)