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Theorem 2rexbiia 3048
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 3042 . 2 (𝑥𝐴 → (∃𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 𝜓))
32rexbiia 3033 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wcel 1987  wrex 2908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-rex 2913
This theorem is referenced by:  cnref1o  11771  mndpfo  17235  mdsymlem8  29118  xlt2addrd  29367  elunirnmbfm  30096  icoreelrnab  32834
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