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Theorem r19.32vdc 2476
 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 2413 . . 3 (∀𝑥𝐴𝜑𝜓) ↔ (¬ 𝜑 → ∀𝑥𝐴 𝜓))
21a1i 9 . 2 (DECID 𝜑 → (∀𝑥𝐴𝜑𝜓) ↔ (¬ 𝜑 → ∀𝑥𝐴 𝜓)))
3 dfordc 802 . . 3 (DECID 𝜑 → ((𝜑𝜓) ↔ (¬ 𝜑𝜓)))
43ralbidv 2343 . 2 (DECID 𝜑 → (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴𝜑𝜓)))
5 dfordc 802 . 2 (DECID 𝜑 → ((𝜑 ∨ ∀𝑥𝐴 𝜓) ↔ (¬ 𝜑 → ∀𝑥𝐴 𝜓)))
62, 4, 53bitr4d 213 1 (DECID 𝜑 → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝐴 𝜓)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 102   ∨ wo 639  DECID wdc 753  ∀wral 2323 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-4 1416  ax-17 1435  ax-ial 1443  ax-i5r 1444 This theorem depends on definitions:  df-bi 114  df-dc 754  df-nf 1366  df-ral 2328 This theorem is referenced by: (None)
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