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Theorem 2rexbidva 2459
Description: Formula-building rule for restricted existential quantifiers (deduction form). (Contributed by NM, 15-Dec-2004.)
Hypothesis
Ref Expression
2ralbidva.1 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))
Assertion
Ref Expression
2rexbidva (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴𝑦𝐵 𝜒))
Distinct variable groups:   𝑥,𝑦,𝜑   𝑦,𝐴
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem 2rexbidva
StepHypRef Expression
1 2ralbidva.1 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))
21anassrs 398 . . 3 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → (𝜓𝜒))
32rexbidva 2435 . 2 ((𝜑𝑥𝐴) → (∃𝑦𝐵 𝜓 ↔ ∃𝑦𝐵 𝜒))
43rexbidva 2435 1 (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴𝑦𝐵 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wcel 1481  wrex 2418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-rex 2423
This theorem is referenced by: (None)
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