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

Theorem r2exf 3317
 Description: Double restricted existential quantification. (Contributed by Mario Carneiro, 14-Oct-2016.) Use r2exlem 3294. (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 3216 . 2 (∀𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ¬ 𝜑))
32r2exlem 3294 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399  ∃wex 1781   ∈ wcel 2115  Ⅎwnfc 2962  ∃wrex 3134 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-12 2179 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139 This theorem is referenced by:  rexcomf  3349
 Copyright terms: Public domain W3C validator