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Theorem 2ralbidva 2363
Description: Formula-building rule for restricted universal quantifiers (deduction rule). (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 1437 . 2 𝑥𝜑
2 nfv 1437 . 2 𝑦𝜑
3 2ralbidva.1 . 2 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))
41, 2, 32ralbida 2362 1 (𝜑 → (∀𝑥𝐴𝑦𝐵 𝜓 ↔ ∀𝑥𝐴𝑦𝐵 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wcel 1409  wral 2323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-4 1416  ax-17 1435
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-ral 2328
This theorem is referenced by:  soinxp  4438  isotr  5484
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