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Theorem r2exf 3270
 Description: Double restricted existential quantification. (Contributed by Mario Carneiro, 14-Oct-2016.) Use r2exlem 3269. (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 3147 . 2 (∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ¬ 𝜑))
32r2exlem 3269 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 198   ∧ wa 386  ∃wex 1880   ∈ wcel 2166  Ⅎwnfc 2956  ∃wrex 3118 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123 This theorem is referenced by:  rexcomf  3307
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