Theorem neneqad 2587

Description: If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2533. One-way deduction form of df-ne 2519. (Contributed by David Moews, 28-Feb-2017.)

Hypothesis

Ref	Expression
neneqad.1	⊢ (φ → ¬ A = B)

Assertion

Ref	Expression
neneqad	⊢ (φ → A ≠ B)

Proof of Theorem neneqad

Step	Hyp	Ref	Expression
1		neneqad.1	. . 3 ⊢ (φ → ¬ A = B)
2	1	con2i 112	. 2 ⊢ (A = B → ¬ φ)
3	2	necon2ai 2562	1 ⊢ (φ → A ≠ B)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1642   ≠ wne 2517

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 2519

This theorem is referenced by: (None)