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Theorem 2ralbida 2362
Description: Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 24-Feb-2004.)
Hypotheses
Ref Expression
2ralbida.1 𝑥𝜑
2ralbida.2 𝑦𝜑
2ralbida.3 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))
Assertion
Ref Expression
2ralbida (𝜑 → (∀𝑥𝐴𝑦𝐵 𝜓 ↔ ∀𝑥𝐴𝑦𝐵 𝜒))
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem 2ralbida
StepHypRef Expression
1 2ralbida.1 . 2 𝑥𝜑
2 2ralbida.2 . . . 4 𝑦𝜑
3 nfv 1437 . . . 4 𝑦 𝑥𝐴
42, 3nfan 1473 . . 3 𝑦(𝜑𝑥𝐴)
5 2ralbida.3 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))
65anassrs 386 . . 3 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → (𝜓𝜒))
74, 6ralbida 2337 . 2 ((𝜑𝑥𝐴) → (∀𝑦𝐵 𝜓 ↔ ∀𝑦𝐵 𝜒))
81, 7ralbida 2337 1 (𝜑 → (∀𝑥𝐴𝑦𝐵 𝜓 ↔ ∀𝑥𝐴𝑦𝐵 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wnf 1365  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:  2ralbidva  2363
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