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Theorem 19.32dc 1566
Description: Theorem 19.32 of [Margaris] p. 90, where φ is decidable. (Contributed by Jim Kingdon, 4-Jun-2018.)
Hypothesis
Ref Expression
19.32dc.1 xφ
Assertion
Ref Expression
19.32dc (DECID φ → (x(φ ψ) ↔ (φ xψ)))

Proof of Theorem 19.32dc
StepHypRef Expression
1 19.32dc.1 . . . . 5 xφ
21nfn 1545 . . . 4 x ¬ φ
3219.21 1472 . . 3 (xφψ) ↔ (¬ φxψ))
43a1i 9 . 2 (DECID φ → (xφψ) ↔ (¬ φxψ)))
51nfdc 1546 . . 3 xDECID φ
6 dfordc 790 . . 3 (DECID φ → ((φ ψ) ↔ (¬ φψ)))
75, 6albid 1503 . 2 (DECID φ → (x(φ ψ) ↔ xφψ)))
8 dfordc 790 . 2 (DECID φ → ((φ xψ) ↔ (¬ φxψ)))
94, 7, 83bitr4d 209 1 (DECID φ → (x(φ ψ) ↔ (φ xψ)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98   wo 628  DECID wdc 741  wal 1240  wnf 1346
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  ax-4 1397  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-dc 742  df-tru 1245  df-fal 1248  df-nf 1347
This theorem is referenced by: (None)
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