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Theorem 2ralbii 2349
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 2347 . 2 (∀𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 𝜓)
32ralbii 2347 1 (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝐴𝑦𝐵 𝜓)
Colors of variables: wff set class
Syntax hints:  wb 102  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-tru 1262  df-nf 1366  df-ral 2328
This theorem is referenced by:  ordsoexmid  4314  cnvsom  4889  fununi  4995  tpossym  5922
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