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Theorem 19.31r 1568
 Description: One direction of Theorem 19.31 of [Margaris] p. 90. The converse holds in classical logic, but not intuitionistic logic. (Contributed by Jim Kingdon, 28-Jul-2018.)
Hypothesis
Ref Expression
19.31r.1 xψ
Assertion
Ref Expression
19.31r ((xφ ψ) → x(φ ψ))

Proof of Theorem 19.31r
StepHypRef Expression
1 19.31r.1 . . 3 xψ
2119.32r 1567 . 2 ((ψ xφ) → x(ψ φ))
3 orcom 646 . 2 ((xφ ψ) ↔ (ψ xφ))
4 orcom 646 . . 3 ((φ ψ) ↔ (ψ φ))
54albii 1356 . 2 (x(φ ψ) ↔ x(ψ φ))
62, 3, 53imtr4i 190 1 ((xφ ψ) → x(φ ψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 628  ∀wal 1240  Ⅎwnf 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: (None)
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