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Theorem rexim 2526
Description: Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994.) (Proof shortened by Andrew Salmon, 30-May-2011.)
Assertion
Ref Expression
rexim (∀𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))

Proof of Theorem rexim
StepHypRef Expression
1 df-ral 2421 . . . 4 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
2 simpl 108 . . . . . . 7 ((𝑥𝐴𝜑) → 𝑥𝐴)
32a1i 9 . . . . . 6 ((𝑥𝐴 → (𝜑𝜓)) → ((𝑥𝐴𝜑) → 𝑥𝐴))
4 pm3.31 260 . . . . . 6 ((𝑥𝐴 → (𝜑𝜓)) → ((𝑥𝐴𝜑) → 𝜓))
53, 4jcad 305 . . . . 5 ((𝑥𝐴 → (𝜑𝜓)) → ((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
65alimi 1431 . . . 4 (∀𝑥(𝑥𝐴 → (𝜑𝜓)) → ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
71, 6sylbi 120 . . 3 (∀𝑥𝐴 (𝜑𝜓) → ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
8 exim 1578 . . 3 (∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)) → (∃𝑥(𝑥𝐴𝜑) → ∃𝑥(𝑥𝐴𝜓)))
97, 8syl 14 . 2 (∀𝑥𝐴 (𝜑𝜓) → (∃𝑥(𝑥𝐴𝜑) → ∃𝑥(𝑥𝐴𝜓)))
10 df-rex 2422 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
11 df-rex 2422 . 2 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
129, 10, 113imtr4g 204 1 (∀𝑥𝐴 (𝜑𝜓) → (∃𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1329  wex 1468  wcel 1480  wral 2416  wrex 2417
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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-ral 2421  df-rex 2422
This theorem is referenced by:  reximia  2527  reximdai  2530  r19.29  2569  reupick2  3362  ss2iun  3828  chfnrn  5531
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