Theorem necon2abid 2999

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 318	. 2 ⊢ (𝜓 ↔ ¬ ¬ 𝜓)
2		necon2abid.1	. . 3 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜓))
3	2	necon3abid 2993	. 2 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ ¬ ¬ 𝜓))
4	1, 3	bitr4id 293	1 ⊢ (𝜑 → (𝜓 ↔ 𝐴 ≠ 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1570   ≠ wne 2957

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-ne 2958

This theorem is used by:  sossfld  6183  funeldmb  7366  fin23lem24  10328  isf32lem4  10362  sqgt0sr  11119  leltne  11327  xrleltne  13200  xrltne  13218  ge0nemnf  13229  xlt2add  13316  supxrbnd  13384  supxrre2  13387  ioopnfsup  13929  icopnfsup  13930  xblpnfps  24627  xblpnf  24628  nmoreltpnf  31258  nmopreltpnf  32358  mh-inf3f1  37168  elprneb  47925