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Theorem r19.32vdc 2626
Description: Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, where 𝜑 is decidable. (Contributed by Jim Kingdon, 4-Jun-2018.)
Assertion
Ref Expression
r19.32vdc (DECID 𝜑 → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝐴 𝜓)))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem r19.32vdc
StepHypRef Expression
1 r19.21v 2554 . . 3 (∀𝑥𝐴𝜑𝜓) ↔ (¬ 𝜑 → ∀𝑥𝐴 𝜓))
21a1i 9 . 2 (DECID 𝜑 → (∀𝑥𝐴𝜑𝜓) ↔ (¬ 𝜑 → ∀𝑥𝐴 𝜓)))
3 dfordc 892 . . 3 (DECID 𝜑 → ((𝜑𝜓) ↔ (¬ 𝜑𝜓)))
43ralbidv 2477 . 2 (DECID 𝜑 → (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴𝜑𝜓)))
5 dfordc 892 . 2 (DECID 𝜑 → ((𝜑 ∨ ∀𝑥𝐴 𝜓) ↔ (¬ 𝜑 → ∀𝑥𝐴 𝜓)))
62, 4, 53bitr4d 220 1 (DECID 𝜑 → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝐴 𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 105  wo 708  DECID wdc 834  wral 2455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-gen 1449  ax-4 1510  ax-17 1526  ax-ial 1534  ax-i5r 1535
This theorem depends on definitions:  df-bi 117  df-dc 835  df-nf 1461  df-ral 2460
This theorem is referenced by: (None)
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