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Theorem 19.32dc 1585
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(𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓)))

Proof of Theorem 19.32dc
StepHypRef Expression
1 19.32dc.1 . . . . 5 𝑥𝜑
21nfn 1564 . . . 4 𝑥 ¬ 𝜑
3219.21 1491 . . 3 (∀𝑥𝜑𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓))
43a1i 9 . 2 (DECID 𝜑 → (∀𝑥𝜑𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓)))
51nfdc 1565 . . 3 𝑥DECID 𝜑
6 dfordc 802 . . 3 (DECID 𝜑 → ((𝜑𝜓) ↔ (¬ 𝜑𝜓)))
75, 6albid 1522 . 2 (DECID 𝜑 → (∀𝑥(𝜑𝜓) ↔ ∀𝑥𝜑𝜓)))
8 dfordc 802 . 2 (DECID 𝜑 → ((𝜑 ∨ ∀𝑥𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓)))
94, 7, 83bitr4d 213 1 (DECID 𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 102  wo 639  DECID wdc 753  wal 1257  wnf 1365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-gen 1354  ax-ie2 1399  ax-4 1416  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-dc 754  df-tru 1262  df-fal 1265  df-nf 1366
This theorem is referenced by: (None)
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