Theorem necon3bii 2549

Description: Inference from equality to inequality. (Contributed by NM, 23-Feb-2005.)

Hypothesis

Ref	Expression
necon3bii.1	⊢ (A = B ↔ C = D)

Assertion

Ref	Expression
necon3bii	⊢ (A ≠ B ↔ C ≠ D)

Proof of Theorem necon3bii

Step	Hyp	Ref	Expression
1		necon3bii.1	. . 3 ⊢ (A = B ↔ C = D)
2	1	necon3abii 2547	. 2 ⊢ (A ≠ B ↔ ¬ C = D)
3		df-ne 2519	. 2 ⊢ (C ≠ D ↔ ¬ C = D)
4	2, 3	bitr4i 243	1 ⊢ (A ≠ B ↔ C ≠ D)

Syntax hints:  ¬ wn 3   ↔ wb 176   = 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:  necom  2598  nulnnn  4557  rnsnn0  5066  ce2  6193  nchoicelem14  6303