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Theorem r19.36v 3327
Description: Restricted quantifier version of one direction of 19.36 2233. (The other direction holds iff 𝐴 is nonempty, see r19.36zv 4425.) (Contributed by NM, 22-Oct-2003.)
Assertion
Ref Expression
r19.36v (∃𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑𝜓))
Distinct variable group:   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem r19.36v
StepHypRef Expression
1 r19.35 3326 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
2 id 22 . . . 4 (𝜓𝜓)
32rexlimivw 3268 . . 3 (∃𝑥𝐴 𝜓𝜓)
43imim2i 16 . 2 ((∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓) → (∀𝑥𝐴 𝜑𝜓))
51, 4sylbi 220 1 (∃𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wral 3126  wrex 3127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-ral 3131  df-rex 3132
This theorem is referenced by:  iinss  4953  uniimadom  9943  hashgt12el  13767
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