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Theorem r2exf 3279
Description: Double restricted existential quantification. For a version based on fewer axioms see r2ex 3195. (Contributed by Mario Carneiro, 14-Oct-2016.) Use r2exlem 3143. (Revised by Wolf Lammen, 10-Jan-2020.)
Hypothesis
Ref Expression
r2exf.1 𝑦𝐴
Assertion
Ref Expression
r2exf (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem r2exf
StepHypRef Expression
1 r2exf.1 . . 3 𝑦𝐴
21r2alf 3278 . 2 (∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ¬ 𝜑))
32r2exlem 3143 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 396  wex 1781  wcel 2106  wnfc 2883  wrex 3070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-12 2171
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-nf 1786  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071
This theorem is referenced by:  rexcomf  3300
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