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Theorem 19.41h 1661
 Description: Theorem 19.41 of [Margaris] p. 90. New proofs should use 19.41 1662 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 1607 . . 3 (∃𝑥(𝜑𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
2 19.41h.1 . . . . 5 (𝜓 → ∀𝑥𝜓)
3 id 19 . . . . 5 (𝜓𝜓)
42, 3exlimih 1569 . . . 4 (∃𝑥𝜓𝜓)
54anim2i 340 . . 3 ((∃𝑥𝜑 ∧ ∃𝑥𝜓) → (∃𝑥𝜑𝜓))
61, 5syl 14 . 2 (∃𝑥(𝜑𝜓) → (∃𝑥𝜑𝜓))
7 pm3.21 262 . . . 4 (𝜓 → (𝜑 → (𝜑𝜓)))
82, 7eximdh 1587 . . 3 (𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))
98impcom 124 . 2 ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓))
106, 9impbii 125 1 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1330  ∃wex 1469 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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1487  ax-ial 1511 This theorem depends on definitions:  df-bi 116 This theorem is referenced by:  19.42h  1663  sbh  1745  sbidm  1819  19.41v  1870  2exeu  2082
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