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Theorem r19.37zv 4432
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 3271 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
2 r19.3rzv 4429 . . 3 (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥𝐴 𝜑))
32imbi1d 342 . 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 205  wne 2943  wral 3064  wrex 3065  c0 4256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-ne 2944  df-ral 3069  df-rex 3070  df-dif 3890  df-nul 4257
This theorem is referenced by:  ishlat3N  37368  hlsupr2  37401
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