Theorem necon3bbid 2551

Description: Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.)

Hypothesis

Ref	Expression
necon3bbid.1	⊢ (φ → (ψ ↔ A = B))

Assertion

Ref	Expression
necon3bbid	⊢ (φ → (¬ ψ ↔ A ≠ B))

Proof of Theorem necon3bbid

Step	Hyp	Ref	Expression
1		necon3bbid.1	. . . 4 ⊢ (φ → (ψ ↔ A = B))
2	1	bicomd 192	. . 3 ⊢ (φ → (A = B ↔ ψ))
3	2	necon3abid 2550	. 2 ⊢ (φ → (A ≠ B ↔ ¬ ψ))
4	3	bicomd 192	1 ⊢ (φ → (¬ ψ ↔ A ≠ B))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   = wceq 1642   ≠ wne 2517

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-ne 2519

This theorem is used by:  necon3bid  2552  eldifsn  3840  brltc  6115  addceq0  6220