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Theorem r19.41 3119
Description: Restricted quantifier version of 19.41 2141. See r19.41v 3118 for a version with a dv condition, requiring fewer axioms. (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 682 . . . 4 (((𝑥𝐴𝜑) ∧ 𝜓) ↔ (𝑥𝐴 ∧ (𝜑𝜓)))
21exbii 1814 . . 3 (∃𝑥((𝑥𝐴𝜑) ∧ 𝜓) ↔ ∃𝑥(𝑥𝐴 ∧ (𝜑𝜓)))
3 r19.41.1 . . . 4 𝑥𝜓
4319.41 2141 . . 3 (∃𝑥((𝑥𝐴𝜑) ∧ 𝜓) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ 𝜓))
52, 4bitr3i 266 . 2 (∃𝑥(𝑥𝐴 ∧ (𝜑𝜓)) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ 𝜓))
6 df-rex 2947 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ ∃𝑥(𝑥𝐴 ∧ (𝜑𝜓)))
7 df-rex 2947 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
87anbi1i 731 . 2 ((∃𝑥𝐴 𝜑𝜓) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ 𝜓))
95, 6, 83bitr4i 292 1 (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wex 1744  wnf 1748  wcel 2030  wrex 2942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-nf 1750  df-rex 2947
This theorem is referenced by:  iunin1f  29500
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