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Theorem 19.32r 1552
Description: One direction of Theorem 19.32 of [Margaris] p. 90. The converse holds if φ is decidable, as seen at 19.32dc 1551. (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 620 . . 3 (φ → (φ ψ))
31, 2alrimi 1396 . 2 (φx(φ ψ))
4 olc 619 . . 3 (ψ → (φ ψ))
54alimi 1324 . 2 (xψx(φ ψ))
63, 5jaoi 623 1 ((φ xψ) → x(φ ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wo 616  wal 1226  wnf 1329
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 617  ax-5 1316  ax-gen 1318  ax-4 1381
This theorem depends on definitions:  df-bi 110  df-nf 1330
This theorem is referenced by:  19.31r  1553
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