Theorem necon3abii 2241

Description: Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007.)

Hypothesis

Ref	Expression
necon3abii.1	⊢ (𝐴 = 𝐵 ↔ 𝜑)

Assertion

Ref	Expression
necon3abii	⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝜑)

Proof of Theorem necon3abii

Step	Hyp	Ref	Expression
1		df-ne 2206	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		necon3abii.1	. 2 ⊢ (𝐴 = 𝐵 ↔ 𝜑)
3	1, 2	xchbinx 607	1 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 98   = wceq 1243   ≠ wne 2204

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545

This theorem depends on definitions:  df-bi 110  df-ne 2206

This theorem is referenced by:  necon3bbii  2242  necon3bii  2243  nesym  2250  n0rf  3233