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Theorem 2rexbiia 3298
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 3296 . 2 (𝑥𝐴 → (∃𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 𝜓))
32rexbiia 3246 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wcel 2110  wrex 3139
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 1907
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-rex 3144
This theorem is referenced by:  reu3op  6142  opreu2reurex  6144  cnref1o  12383  mndpfo  17933  wspthsnwspthsnon  27694  mdsymlem8  30186  xlt2addrd  30481  elunirnmbfm  31511  satfv0  32605  fmla0xp  32630  icoreelrnab  34634  rrx2xpref1o  44704
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