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Theorem 19.41h 1685
Description: Theorem 19.41 of [Margaris] p. 90. New proofs should use 19.41 1686 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
19.41h.1 (𝜓 → ∀𝑥𝜓)
Assertion
Ref Expression
19.41h (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Proof of Theorem 19.41h
StepHypRef Expression
1 19.40 1631 . . 3 (∃𝑥(𝜑𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
2 19.41h.1 . . . . 5 (𝜓 → ∀𝑥𝜓)
3 id 19 . . . . 5 (𝜓𝜓)
42, 3exlimih 1593 . . . 4 (∃𝑥𝜓𝜓)
54anim2i 342 . . 3 ((∃𝑥𝜑 ∧ ∃𝑥𝜓) → (∃𝑥𝜑𝜓))
61, 5syl 14 . 2 (∃𝑥(𝜑𝜓) → (∃𝑥𝜑𝜓))
7 pm3.21 264 . . . 4 (𝜓 → (𝜑 → (𝜑𝜓)))
82, 7eximdh 1611 . . 3 (𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))
98impcom 125 . 2 ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓))
106, 9impbii 126 1 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1351  wex 1492
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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  19.42h  1687  sbh  1776  sbidm  1851  19.41v  1902  2exeu  2118
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