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Theorem r19.42v 2621
Description: Restricted version of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
Assertion
Ref Expression
r19.42v (∃𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝐴 𝜓))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem r19.42v
StepHypRef Expression
1 r19.41v 2620 . 2 (∃𝑥𝐴 (𝜓𝜑) ↔ (∃𝑥𝐴 𝜓𝜑))
2 ancom 264 . . 3 ((𝜑𝜓) ↔ (𝜓𝜑))
32rexbii 2471 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ ∃𝑥𝐴 (𝜓𝜑))
4 ancom 264 . 2 ((𝜑 ∧ ∃𝑥𝐴 𝜓) ↔ (∃𝑥𝐴 𝜓𝜑))
51, 3, 43bitr4i 211 1 (∃𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝐴 𝜓))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wrex 2443
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 1434  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-4 1497  ax-17 1513  ax-ial 1521
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-rex 2448
This theorem is referenced by:  ceqsrexbv  2853  ceqsrex2v  2854  2reuswapdc  2926  iunrab  3908  iunin2  3924  iundif2ss  3926  iunopab  4254  elxp2  4617  cnvuni  4785  elunirn  5729  f1oiso  5789  oprabrexex2  6091  genpdflem  7440  1idprl  7523  1idpru  7524  ltexprlemm  7533  rexuz2  9511  4fvwrd4  10066  divalgb  11848
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