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Theorem r19.37zv 4209
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 r19.3rzv 4206 . . 3 (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥𝐴 𝜑))
21imbi1d 330 . 2 (𝐴 ≠ ∅ → ((𝜑 → ∃𝑥𝐴 𝜓) ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓)))
3 r19.35 3232 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
42, 3syl6rbbr 279 1 (𝐴 ≠ ∅ → (∃𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∃𝑥𝐴 𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wne 2943  wral 3061  wrex 3062  c0 4063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-v 3353  df-dif 3726  df-nul 4064
This theorem is referenced by:  ishlat3N  35163  hlsupr2  35196
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