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Theorem r19.40 2481
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 106 . . 3 ((𝜑𝜓) → 𝜑)
21reximi 2433 . 2 (∃𝑥𝐴 (𝜑𝜓) → ∃𝑥𝐴 𝜑)
3 simpr 107 . . 3 ((𝜑𝜓) → 𝜓)
43reximi 2433 . 2 (∃𝑥𝐴 (𝜑𝜓) → ∃𝑥𝐴 𝜓)
52, 4jca 294 1 (∃𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-ral 2328  df-rex 2329
This theorem is referenced by:  rexanuz  9815
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