ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  19.30dc Structured version   GIF version

Theorem 19.30dc 1515
Description: Theorem 19.30 of [Margaris] p. 90, with an additional decidability condition. (Contributed by Jim Kingdon, 21-Jul-2018.)
Assertion
Ref Expression
19.30dc (DECID xψ → (x(φ ψ) → (xφ xψ)))

Proof of Theorem 19.30dc
StepHypRef Expression
1 df-dc 742 . 2 (DECID xψ ↔ (xψ ¬ xψ))
2 olc 631 . . . 4 (xψ → (xφ xψ))
32a1d 22 . . 3 (xψ → (x(φ ψ) → (xφ xψ)))
4 alnex 1385 . . . . 5 (x ¬ ψ ↔ ¬ xψ)
5 orel2 644 . . . . . 6 ψ → ((φ ψ) → φ))
65al2imi 1344 . . . . 5 (x ¬ ψ → (x(φ ψ) → xφ))
74, 6sylbir 125 . . . 4 xψ → (x(φ ψ) → xφ))
8 orc 632 . . . 4 (xφ → (xφ xψ))
97, 8syl6 29 . . 3 xψ → (x(φ ψ) → (xφ xψ)))
103, 9jaoi 635 . 2 ((xψ ¬ xψ) → (x(φ ψ) → (xφ xψ)))
111, 10sylbi 114 1 (DECID xψ → (x(φ ψ) → (xφ xψ)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wo 628  DECID wdc 741  wal 1240  wex 1378
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-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-gen 1335  ax-ie2 1380
This theorem depends on definitions:  df-bi 110  df-dc 742  df-tru 1245  df-fal 1248
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator