Theorem necon3bbii 2547

Description: Deduction from equality to inequality. (Contributed by NM, 13-Apr-2007.)

Hypothesis

Ref	Expression
necon3bbii.1	⊢ (φ ↔ A = B)

Assertion

Ref	Expression
necon3bbii	⊢ (¬ φ ↔ A ≠ B)

Proof of Theorem necon3bbii

Step	Hyp	Ref	Expression
1		necon3bbii.1	. . . 4 ⊢ (φ ↔ A = B)
2	1	bicomi 193	. . 3 ⊢ (A = B ↔ φ)
3	2	necon3abii 2546	. 2 ⊢ (A ≠ B ↔ ¬ φ)
4	3	bicomi 193	1 ⊢ (¬ φ ↔ A ≠ B)

Syntax hints:  ¬ wn 3   ↔ wb 176   = 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:  nssinpss  3487  difsnpss  3851  foundex  5914  ce0nn  6180