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Theorem r19.40 3350
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 483 . . 3 ((𝜑𝜓) → 𝜑)
21reximi 3247 . 2 (∃𝑥𝐴 (𝜑𝜓) → ∃𝑥𝐴 𝜑)
3 simpr 485 . . 3 ((𝜑𝜓) → 𝜓)
43reximi 3247 . 2 (∃𝑥𝐴 (𝜑𝜓) → ∃𝑥𝐴 𝜓)
52, 4jca 512 1 (∃𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wrex 3143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-ral 3147  df-rex 3148
This theorem is referenced by:  rexanuz  14698  txflf  22530  metequiv2  23035  mzpcompact2lem  39210
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