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Theorem 2ralbidv 2403
Description: Formula-building rule for restricted universal quantifiers (deduction form). (Contributed by NM, 28-Jan-2006.) (Revised by Szymon Jaroszewicz, 16-Mar-2007.)
Hypothesis
Ref Expression
2ralbidv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
2ralbidv (𝜑 → (∀𝑥𝐴𝑦𝐵 𝜓 ↔ ∀𝑥𝐴𝑦𝐵 𝜒))
Distinct variable groups:   𝜑,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem 2ralbidv
StepHypRef Expression
1 2ralbidv.1 . . 3 (𝜑 → (𝜓𝜒))
21ralbidv 2381 . 2 (𝜑 → (∀𝑦𝐵 𝜓 ↔ ∀𝑦𝐵 𝜒))
32ralbidv 2381 1 (𝜑 → (∀𝑥𝐴𝑦𝐵 𝜓 ↔ ∀𝑥𝐴𝑦𝐵 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wral 2360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-4 1446  ax-17 1465
This theorem depends on definitions:  df-bi 116  df-nf 1396  df-ral 2365
This theorem is referenced by:  cbvral3v  2601  poeq1  4135  soeq1  4151  isoeq1  5594  isoeq2  5595  isoeq3  5596  smoeq  6069  xpf1o  6614  elinp  7094  cauappcvgpr  7282  iseqcaopr2  9972  addcn2  10760  mulcn2  10762  elcncf  11902
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