MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r2exf Structured version   Visualization version   GIF version

Theorem r2exf 3276
Description: Double restricted existential quantification. For a version based on fewer axioms see r2ex 3192. (Contributed by Mario Carneiro, 14-Oct-2016.) Use r2exlem 3140. (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 3275 . 2 (∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ¬ 𝜑))
32r2exlem 3140 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 395  wex 1774  wcel 2099  wnfc 2879  wrex 3067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-12 2167
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-nf 1779  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068
This theorem is referenced by:  rexcomf  3297
  Copyright terms: Public domain W3C validator