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Theorem r19.41 2612
Description: Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 1-Nov-2010.)
Hypothesis
Ref Expression
r19.41.1 𝑥𝜓
Assertion
Ref Expression
r19.41 (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))

Proof of Theorem r19.41
StepHypRef Expression
1 anass 399 . . . 4 (((𝑥𝐴𝜑) ∧ 𝜓) ↔ (𝑥𝐴 ∧ (𝜑𝜓)))
21exbii 1585 . . 3 (∃𝑥((𝑥𝐴𝜑) ∧ 𝜓) ↔ ∃𝑥(𝑥𝐴 ∧ (𝜑𝜓)))
3 r19.41.1 . . . 4 𝑥𝜓
4319.41 1666 . . 3 (∃𝑥((𝑥𝐴𝜑) ∧ 𝜓) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ 𝜓))
52, 4bitr3i 185 . 2 (∃𝑥(𝑥𝐴 ∧ (𝜑𝜓)) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ 𝜓))
6 df-rex 2441 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ ∃𝑥(𝑥𝐴 ∧ (𝜑𝜓)))
7 df-rex 2441 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
87anbi1i 454 . 2 ((∃𝑥𝐴 𝜑𝜓) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ 𝜓))
95, 6, 83bitr4i 211 1 (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wnf 1440  wex 1472  wcel 2128  wrex 2436
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 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-rex 2441
This theorem is referenced by:  r19.41v  2613
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