Theorem necon2abid 2968

Description: Contrapositive deduction for inequality. (Contributed by NM, 18-Jul-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)

Hypothesis

Ref	Expression
necon2abid.1	⊢ (𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜓))

Assertion

Ref	Expression
necon2abid	⊢ (𝜑 → (𝜓 ↔ 𝐴 ≠ 𝐵))

Proof of Theorem necon2abid

Step	Hyp	Ref	Expression
1		notnotb 315	. 2 ⊢ (𝜓 ↔ ¬ ¬ 𝜓)
2		necon2abid.1	. . 3 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜓))
3	2	necon3abid 2962	. 2 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ ¬ ¬ 𝜓))
4	1, 3	bitr4id 290	1 ⊢ (𝜑 → (𝜓 ↔ 𝐴 ≠ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1540   ≠ wne 2926

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 207  df-ne 2927

This theorem is referenced by:  sossfld  6162  funeldmb  7337  fin23lem24  10282  isf32lem4  10316  sqgt0sr  11066  leltne  11270  xrleltne  13112  xrltne  13130  ge0nemnf  13140  xlt2add  13227  supxrbnd  13295  supxrre2  13298  ioopnfsup  13833  icopnfsup  13834  xblpnfps  24290  xblpnf  24291  nmoreltpnf  30705  nmopreltpnf  31805  elprneb  47034