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Theorem 2ralbidva 2455
Description: Formula-building rule for restricted universal quantifiers (deduction form). (Contributed by NM, 4-Mar-1997.)
Hypothesis
Ref Expression
2ralbidva.1 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))
Assertion
Ref Expression
2ralbidva (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 2ralbidva
StepHypRef Expression
1 nfv 1508 . 2 𝑥𝜑
2 nfv 1508 . 2 𝑦𝜑
3 2ralbidva.1 . 2 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))
41, 2, 32ralbida 2454 1 (𝜑 → (∀𝑥𝐴𝑦𝐵 𝜓 ↔ ∀𝑥𝐴𝑦𝐵 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wcel 1480  wral 2414
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 1423  ax-gen 1425  ax-4 1487  ax-17 1506
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-ral 2419
This theorem is referenced by:  soinxp  4604  isotr  5710  fnmpoovd  6105  ismet2  12512  txmetcn  12677
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