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Theorem 2rexbii 2499
Description: Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 11-Nov-1995.)
Hypothesis
Ref Expression
ralbii.1 (𝜑𝜓)
Assertion
Ref Expression
2rexbii (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜓)

Proof of Theorem 2rexbii
StepHypRef Expression
1 ralbii.1 . . 3 (𝜑𝜓)
21rexbii 2497 . 2 (∃𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 𝜓)
32rexbii 2497 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜓)
Colors of variables: wff set class
Syntax hints:  wb 105  wrex 2469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-rex 2474
This theorem is referenced by:  3reeanv  2661  4fvwrd4  10172  prodmodc  11621  pythagtriplem2  12301  pythagtrip  12318
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