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Theorem r19.37 2456
 Description: Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. In classical logic the converse would hold if A has at least one element, but that is not sufficient in intuitionistic logic. (Contributed by FL, 13-May-2012.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypothesis
Ref Expression
r19.37.1 xφ
Assertion
Ref Expression
r19.37 (x A (φψ) → (φx A ψ))

Proof of Theorem r19.37
StepHypRef Expression
1 r19.37.1 . . 3 xφ
2 ax-1 5 . . 3 (φ → (x Aφ))
31, 2ralrimi 2384 . 2 (φx A φ)
4 r19.35-1 2454 . 2 (x A (φψ) → (x A φx A ψ))
53, 4syl5 28 1 (x A (φψ) → (φx A ψ))
 Colors of variables: wff set class Syntax hints:   → wi 4  Ⅎwnf 1346   ∈ wcel 1390  ∀wral 2300  ∃wrex 2301 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-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-ial 1424 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-ral 2305  df-rex 2306 This theorem is referenced by:  r19.37av  2457
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