Theorem necon2abid 3001

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 2995	. 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 2959

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 2960

This theorem is used by:  sossfld  6191  funeldmb  7395  fin23lem24  10430  isf32lem4  10464  sqgt0sr  11221  leltne  11429  xrleltne  13304  xrltne  13322  ge0nemnf  13333  xlt2add  13420  supxrbnd  13488  supxrre2  13491  ioopnfsup  14034  icopnfsup  14035  xblpnfps  24747  xblpnf  24748  nmoreltpnf  31404  nmopreltpnf  32504  elprneb  48314