Theorem necon3abii 2828

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 2782	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		necon3abii.1	. 2 ⊢ (𝐴 = 𝐵 ↔ 𝜑)
3	1, 2	xchbinx 323	1 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 195   = wceq 1475   ≠ wne 2780

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 196  df-ne 2782

This theorem is referenced by:  necon3bbii  2829  necon3bii  2834  nesym  2838  n0fOLD  3887  rabn0  3912  xpimasn  5498  rankxplim3  8627  rankxpsuc  8628  dflt2  11857  gcd0id  15078  lcmfunsnlem2  15191  axlowdimlem13  25634  filnetlem4  31546  dihatlat  35641  pellex  36417  nev  37081  ldepspr  42056