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Theorem 19.30dc 1496
 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 731 . 2 (DECID xψ ↔ (xψ ¬ xψ))
2 olc 619 . . . 4 (xψ → (xφ xψ))
32a1d 22 . . 3 (xψ → (x(φ ψ) → (xφ xψ)))
4 alnex 1365 . . . . 5 (x ¬ ψ ↔ ¬ xψ)
5 orel2 632 . . . . . 6 ψ → ((φ ψ) → φ))
65al2imi 1323 . . . . 5 (x ¬ ψ → (x(φ ψ) → xφ))
74, 6sylbir 125 . . . 4 xψ → (x(φ ψ) → xφ))
8 orc 620 . . . 4 (xφ → (xφ xψ))
97, 8syl6 29 . . 3 xψ → (x(φ ψ) → (xφ xψ)))
103, 9jaoi 623 . 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 616  DECID wdc 730  ∀wal 1224  ∃wex 1358 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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-gen 1314  ax-ie2 1360 This theorem depends on definitions:  df-bi 110  df-dc 731  df-tru 1229  df-fal 1232 This theorem is referenced by: (None)
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