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Theorem r19.37 2479
 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 (∃𝑥𝐴 (𝜑𝜓) → (𝜑 → ∃𝑥𝐴 𝜓))

Proof of Theorem r19.37
StepHypRef Expression
1 r19.37.1 . . 3 𝑥𝜑
2 ax-1 5 . . 3 (𝜑 → (𝑥𝐴𝜑))
31, 2ralrimi 2407 . 2 (𝜑 → ∀𝑥𝐴 𝜑)
4 r19.35-1 2477 . 2 (∃𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
53, 4syl5 32 1 (∃𝑥𝐴 (𝜑𝜓) → (𝜑 → ∃𝑥𝐴 𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4  Ⅎwnf 1365   ∈ wcel 1409  ∀wral 2323  ∃wrex 2324 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443 This theorem depends on definitions:  df-bi 114  df-nf 1366  df-ral 2328  df-rex 2329 This theorem is referenced by:  r19.37av  2480
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