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Theorem 19.41h 1620
Description: Theorem 19.41 of [Margaris] p. 90. New proofs should use 19.41 1621 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 1567 . . 3 (∃𝑥(𝜑𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
2 19.41h.1 . . . . 5 (𝜓 → ∀𝑥𝜓)
3 id 19 . . . . 5 (𝜓𝜓)
42, 3exlimih 1529 . . . 4 (∃𝑥𝜓𝜓)
54anim2i 334 . . 3 ((∃𝑥𝜑 ∧ ∃𝑥𝜓) → (∃𝑥𝜑𝜓))
61, 5syl 14 . 2 (∃𝑥(𝜑𝜓) → (∃𝑥𝜑𝜓))
7 pm3.21 260 . . . 4 (𝜓 → (𝜑 → (𝜑𝜓)))
82, 7eximdh 1547 . . 3 (𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))
98impcom 123 . 2 ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓))
106, 9impbii 124 1 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1287  wex 1426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-ial 1472
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  19.42h  1622  sbh  1706  sbidm  1779  19.41v  1830  2exeu  2040
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