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Theorem r19.40 2562
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 108 . . 3 ((𝜑𝜓) → 𝜑)
21reximi 2506 . 2 (∃𝑥𝐴 (𝜑𝜓) → ∃𝑥𝐴 𝜑)
3 simpr 109 . . 3 ((𝜑𝜓) → 𝜓)
43reximi 2506 . 2 (∃𝑥𝐴 (𝜑𝜓) → ∃𝑥𝐴 𝜓)
52, 4jca 304 1 (∃𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wrex 2394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-ial 1499
This theorem depends on definitions:  df-bi 116  df-ral 2398  df-rex 2399
This theorem is referenced by:  rexanuz  10728  metequiv2  12592
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