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Theorem r19.32vdc 2453
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 → (x A (φ ψ) ↔ (φ x A ψ)))
Distinct variable group:   φ,x
Allowed substitution hints:   ψ(x)   A(x)

Proof of Theorem r19.32vdc
StepHypRef Expression
1 r19.21v 2390 . . 3 (x Aφψ) ↔ (¬ φx A ψ))
21a1i 9 . 2 (DECID φ → (x Aφψ) ↔ (¬ φx A ψ)))
3 dfordc 790 . . 3 (DECID φ → ((φ ψ) ↔ (¬ φψ)))
43ralbidv 2320 . 2 (DECID φ → (x A (φ ψ) ↔ x Aφψ)))
5 dfordc 790 . 2 (DECID φ → ((φ x A ψ) ↔ (¬ φx A ψ)))
62, 4, 53bitr4d 209 1 (DECID φ → (x A (φ ψ) ↔ (φ x A ψ)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98   wo 628  DECID wdc 741  wral 2300
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-4 1397  ax-17 1416  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-dc 742  df-nf 1347  df-ral 2305
This theorem is referenced by: (None)
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