Theorem necon2abid 2983

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 2977	. 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 2940

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 2941

This theorem is referenced by:  sossfld  6206  funeldmb  7379  fin23lem24  10362  isf32lem4  10396  sqgt0sr  11146  leltne  11350  xrleltne  13187  xrltne  13205  ge0nemnf  13215  xlt2add  13302  supxrbnd  13370  supxrre2  13373  ioopnfsup  13904  icopnfsup  13905  xblpnfps  24405  xblpnf  24406  nmoreltpnf  30788  nmopreltpnf  31888  elprneb  47041