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Theorem 19.32r 1567
Description: One direction of Theorem 19.32 of [Margaris] p. 90. The converse holds if φ is decidable, as seen at 19.32dc 1566. (Contributed by Jim Kingdon, 28-Jul-2018.)
Hypothesis
Ref Expression
19.32r.1 xφ
Assertion
Ref Expression
19.32r ((φ xψ) → x(φ ψ))

Proof of Theorem 19.32r
StepHypRef Expression
1 19.32r.1 . . 3 xφ
2 orc 632 . . 3 (φ → (φ ψ))
31, 2alrimi 1412 . 2 (φx(φ ψ))
4 olc 631 . . 3 (ψ → (φ ψ))
54alimi 1341 . 2 (xψx(φ ψ))
63, 5jaoi 635 1 ((φ xψ) → x(φ ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wo 628  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-io 629  ax-5 1333  ax-gen 1335  ax-4 1397
This theorem depends on definitions:  df-bi 110  df-nf 1347
This theorem is referenced by:  19.31r  1568
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