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Theorem r19.41 3261
Description: Restricted quantifier version of 19.41 2233. See r19.41v 3187 for a version with a disjoint variable 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 df-rex 3069 . 2 (∃𝑥𝐴 (𝜑𝜓) ↔ ∃𝑥(𝑥𝐴 ∧ (𝜑𝜓)))
2 anass 468 . . 3 (((𝑥𝐴𝜑) ∧ 𝜓) ↔ (𝑥𝐴 ∧ (𝜑𝜓)))
32exbii 1845 . 2 (∃𝑥((𝑥𝐴𝜑) ∧ 𝜓) ↔ ∃𝑥(𝑥𝐴 ∧ (𝜑𝜓)))
4 r19.41.1 . . . 4 𝑥𝜓
5419.41 2233 . . 3 (∃𝑥((𝑥𝐴𝜑) ∧ 𝜓) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ 𝜓))
6 df-rex 3069 . . . 4 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
76bicomi 224 . . 3 (∃𝑥(𝑥𝐴𝜑) ↔ ∃𝑥𝐴 𝜑)
85, 7bianbi 627 . 2 (∃𝑥((𝑥𝐴𝜑) ∧ 𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))
91, 3, 83bitr2i 299 1 (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wex 1776  wnf 1780  wcel 2106  wrex 3068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-nf 1781  df-rex 3069
This theorem is referenced by:  reuxfrdf  32519  iunin1f  32578  2reu8i  47063
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