Theorem necon3abid 2379

Description: Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.)

Hypothesis

Ref	Expression
necon3abid.1	⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝜓))

Assertion

Ref	Expression
necon3abid	⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ ¬ 𝜓))

Proof of Theorem necon3abid

Step	Hyp	Ref	Expression
1		df-ne 2341	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		necon3abid.1	. . 3 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝜓))
3	2	notbid 662	. 2 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝜓))
4	1, 3	syl5bb 191	1 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   = wceq 1348   ≠ wne 2340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610

This theorem depends on definitions:  df-bi 116  df-ne 2341

This theorem is referenced by:  necon3bbid  2380  fndmdif  5601  expnegap0  10484  gcdn0gt0  11933  cncongr2  12058