Theorem necon3abii 3045

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 3000	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		necon3abii.1	. 2 ⊢ (𝐴 = 𝐵 ↔ 𝜑)
3	1, 2	xchbinx 326	1 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 198   = wceq 1656   ≠ wne 2999

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 199  df-ne 3000

This theorem is referenced by:  necon3bbii  3046  necon3bii  3051  nesym  3055  rabn0  4187  xpimasn  5820  rankxplim3  9021  rankxpsuc  9022  dflt2  12267  gcd0id  15613  lcmfunsnlem2  15726  axlowdimlem13  26253  filnetlem4  32903  dihatlat  37402  pellex  38236  nev  38896  ldepspr  43102