Theorem necon2abid 2988

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 2982	. 2 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ ¬ ¬ 𝜓))
4	1, 3	bitr4id 290	1 ⊢ (𝜑 → (𝜓 ↔ 𝐴 ≠ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   = wceq 1542   ≠ wne 2945

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 206  df-ne 2946

This theorem is referenced by:  sossfld  6088  fin23lem24  10079  isf32lem4  10113  sqgt0sr  10863  leltne  11065  xrleltne  12878  xrltne  12896  ge0nemnf  12906  xlt2add  12993  supxrbnd  13061  supxrre2  13064  ioopnfsup  13582  icopnfsup  13583  xblpnfps  23546  xblpnf  23547  nmoreltpnf  29127  nmopreltpnf  30227  funeldmb  33733  elprneb  44491