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Theorem 2ralbida 3237
 Description: Formula-building rule for restricted universal quantifier (deduction form). (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 1908 . . . 4 𝑦 𝑥𝐴
42, 3nfan 1893 . . 3 𝑦(𝜑𝑥𝐴)
5 2ralbida.3 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))
65anassrs 468 . . 3 (((𝜑𝑥𝐴) ∧ 𝑦𝐵) → (𝜓𝜒))
74, 6ralbida 3235 . 2 ((𝜑𝑥𝐴) → (∀𝑦𝐵 𝜓 ↔ ∀𝑦𝐵 𝜒))
81, 7ralbida 3235 1 (𝜑 → (∀𝑥𝐴𝑦𝐵 𝜓 ↔ ∀𝑥𝐴𝑦𝐵 𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  Ⅎwnf 1777   ∈ wcel 2107  ∀wral 3143 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-12 2169 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-ral 3148 This theorem is referenced by: (None)
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