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Theorem exim 1610
Description: Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)
Assertion
Ref Expression
exim (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))

Proof of Theorem exim
StepHypRef Expression
1 hba1 1551 . 2 (∀𝑥(𝜑𝜓) → ∀𝑥𝑥(𝜑𝜓))
2 hbe1 1506 . 2 (∃𝑥𝜓 → ∀𝑥𝑥𝜓)
3 19.8a 1601 . . . 4 (𝜓 → ∃𝑥𝜓)
43imim2i 12 . . 3 ((𝜑𝜓) → (𝜑 → ∃𝑥𝜓))
54sps 1548 . 2 (∀𝑥(𝜑𝜓) → (𝜑 → ∃𝑥𝜓))
61, 2, 5exlimdh 1607 1 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1362  wex 1503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  eximi  1611  exbi  1615  eximdh  1622  19.29  1631  19.25  1637  alexim  1656  19.23t  1688  spimt  1747  equvini  1769  nfexd  1772  ax10oe  1808  sbcof2  1821  spsbim  1854  nf5-1  2036  mor  2080  rexim  2584  elex22  2767  elex2  2768  vtoclegft  2824  spcimgft  2828  spcimegft  2830  spc2gv  2843  spc3gv  2845  ssoprab2  5953  bj-inf2vnlem1  15200
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