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Theorem r19.40 3069
 Description: Restricted quantifier version of Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 2-Apr-2004.)
Assertion
Ref Expression
r19.40 (∃𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 𝜓))

Proof of Theorem r19.40
StepHypRef Expression
1 simpl 472 . . 3 ((𝜑𝜓) → 𝜑)
21reximi 2994 . 2 (∃𝑥𝐴 (𝜑𝜓) → ∃𝑥𝐴 𝜑)
3 simpr 476 . . 3 ((𝜑𝜓) → 𝜓)
43reximi 2994 . 2 (∃𝑥𝐴 (𝜑𝜓) → ∃𝑥𝐴 𝜓)
52, 4jca 553 1 (∃𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∃wrex 2897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-ral 2901  df-rex 2902 This theorem is referenced by:  rexanuz  13933  txflf  21620  metequiv2  22125  mzpcompact2lem  36332
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