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Theorem r19.36av 2585
Description: One direction of a restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. In classical logic, the converse would hold if 𝐴 has at least one element, but in intuitionistic logic, that is not a sufficient condition. (Contributed by NM, 22-Oct-2003.)
Assertion
Ref Expression
r19.36av (∃𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑𝜓))
Distinct variable group:   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem r19.36av
StepHypRef Expression
1 r19.35-1 2584 . 2 (∃𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
2 idd 21 . . . 4 (𝑥𝐴 → (𝜓𝜓))
32rexlimiv 2546 . . 3 (∃𝑥𝐴 𝜓𝜓)
43imim2i 12 . 2 ((∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓) → (∀𝑥𝐴 𝜑𝜓))
51, 4syl 14 1 (∃𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1481  wral 2417  wrex 2418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515  ax-i5r 1516
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-ral 2422  df-rex 2423
This theorem is referenced by:  iinss  3871
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