Theorem necon2abid 2975

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 2969	. 2 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ ¬ ¬ 𝜓))
4	1, 3	bitr4id 290	1 ⊢ (𝜑 → (𝜓 ↔ 𝐴 ≠ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1542   ≠ wne 2933

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 2934

This theorem is referenced by:  sossfld  6144  funeldmb  7307  fin23lem24  10235  isf32lem4  10269  sqgt0sr  11020  leltne  11226  xrleltne  13087  xrltne  13105  ge0nemnf  13116  xlt2add  13203  supxrbnd  13271  supxrre2  13274  ioopnfsup  13814  icopnfsup  13815  xblpnfps  24370  xblpnf  24371  nmoreltpnf  30855  nmopreltpnf  31955  mh-inf3f1  36739  elprneb  47489