MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2rexbiia Structured version   Visualization version   GIF version

Theorem 2rexbiia 3216
Description: Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)
Hypothesis
Ref Expression
2rexbiia.1 ((𝑥𝐴𝑦𝐵) → (𝜑𝜓))
Assertion
Ref Expression
2rexbiia (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜓)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem 2rexbiia
StepHypRef Expression
1 2rexbiia.1 . . 3 ((𝑥𝐴𝑦𝐵) → (𝜑𝜓))
21rexbidva 3175 . 2 (𝑥𝐴 → (∃𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 𝜓))
32rexbiia 3090 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  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
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-rex 3069
This theorem is referenced by:  reu3op  6314  opreu2reurex  6316  cnref1o  13025  mndpfo  18783  wspthsnwspthsnon  29946  mdsymlem8  32439  xlt2addrd  32769  elunirnmbfm  34233  satfv0  35343  fmla0xp  35368  icoreelrnab  37337  fimgmcyclem  42520  rrx2xpref1o  48568
  Copyright terms: Public domain W3C validator