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Theorem r19.37zv 4501
Description: Restricted quantifier version of Theorem 19.37 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by Paul Chapman, 8-Oct-2007.)
Assertion
Ref Expression
r19.37zv (𝐴 ≠ ∅ → (∃𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∃𝑥𝐴 𝜓)))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem r19.37zv
StepHypRef Expression
1 r19.35 3107 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
2 r19.3rzv 4498 . . 3 (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥𝐴 𝜑))
32imbi1d 341 . 2 (𝐴 ≠ ∅ → ((𝜑 → ∃𝑥𝐴 𝜓) ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓)))
41, 3bitr4id 290 1 (𝐴 ≠ ∅ → (∃𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∃𝑥𝐴 𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wne 2939  wral 3060  wrex 3069  c0 4332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-9 2117  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-ne 2940  df-ral 3061  df-rex 3070  df-dif 3953  df-nul 4333
This theorem is referenced by:  ishlat3N  39356  hlsupr2  39390
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