 Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  necon4bbiddc Structured version   GIF version

Theorem necon4bbiddc 2273
 Description: Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
Hypothesis
Ref Expression
necon4bbiddc.1 (φ → (DECID ψ → (DECID A = B → (¬ ψAB))))
Assertion
Ref Expression
necon4bbiddc (φ → (DECID ψ → (DECID A = B → (ψA = B))))

Proof of Theorem necon4bbiddc
StepHypRef Expression
1 necon4bbiddc.1 . . . . . 6 (φ → (DECID ψ → (DECID A = B → (¬ ψAB))))
2 bicom 128 . . . . . 6 ((¬ ψAB) ↔ (AB ↔ ¬ ψ))
31, 2syl8ib 155 . . . . 5 (φ → (DECID ψ → (DECID A = B → (AB ↔ ¬ ψ))))
43com23 72 . . . 4 (φ → (DECID A = B → (DECID ψ → (AB ↔ ¬ ψ))))
54necon4abiddc 2272 . . 3 (φ → (DECID A = B → (DECID ψ → (A = Bψ))))
65com23 72 . 2 (φ → (DECID ψ → (DECID A = B → (A = Bψ))))
7 bicom 128 . 2 ((A = Bψ) ↔ (ψA = B))
86, 7syl8ib 155 1 (φ → (DECID ψ → (DECID A = B → (ψA = B))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98  DECID wdc 741   = wceq 1242   ≠ wne 2201 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 629 This theorem depends on definitions:  df-bi 110  df-dc 742  df-ne 2203 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator