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Theorem 2r19.29 3293
 Description: Theorem r19.29 3219 with two quantifiers. (Contributed by Rodolfo Medina, 25-Sep-2010.)
Assertion
Ref Expression
2r19.29 ((∀𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑥𝐴𝑦𝐵 𝜓) → ∃𝑥𝐴𝑦𝐵 (𝜑𝜓))

Proof of Theorem 2r19.29
StepHypRef Expression
1 r19.29 3219 . 2 ((∀𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑥𝐴𝑦𝐵 𝜓) → ∃𝑥𝐴 (∀𝑦𝐵 𝜑 ∧ ∃𝑦𝐵 𝜓))
2 r19.29 3219 . . 3 ((∀𝑦𝐵 𝜑 ∧ ∃𝑦𝐵 𝜓) → ∃𝑦𝐵 (𝜑𝜓))
32reximi 3209 . 2 (∃𝑥𝐴 (∀𝑦𝐵 𝜑 ∧ ∃𝑦𝐵 𝜓) → ∃𝑥𝐴𝑦𝐵 (𝜑𝜓))
41, 3syl 17 1 ((∀𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑥𝐴𝑦𝐵 𝜓) → ∃𝑥𝐴𝑦𝐵 (𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wral 3109  ∃wrex 3110 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-ral 3114  df-rex 3115 This theorem is referenced by:  rnmposs  30440  prter2  36170
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