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Theorem 2ralbii 2485
Description: Inference adding two restricted universal quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)
Hypothesis
Ref Expression
ralbii.1 (𝜑𝜓)
Assertion
Ref Expression
2ralbii (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝐴𝑦𝐵 𝜓)

Proof of Theorem 2ralbii
StepHypRef Expression
1 ralbii.1 . . 3 (𝜑𝜓)
21ralbii 2483 . 2 (∀𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 𝜓)
32ralbii 2483 1 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝐴𝑦𝐵 𝜓)
Colors of variables: wff set class
Syntax hints:  wb 105  wral 2455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-4 1510  ax-17 1526
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-ral 2460
This theorem is referenced by:  rmo4f  2935  ordsoexmid  4557  cnvsom  5167  fununi  5279  tpossym  6270  axpre-suploc  7879  issubm  12740  isbasis2g  13176
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