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Theorem r19.26-2 2498
Description: Theorem 19.26 of [Margaris] p. 90 with 2 restricted quantifiers. (Contributed by NM, 10-Aug-2004.)
Assertion
Ref Expression
r19.26-2 (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴𝑦𝐵 𝜑 ∧ ∀𝑥𝐴𝑦𝐵 𝜓))

Proof of Theorem r19.26-2
StepHypRef Expression
1 r19.26 2497 . . 3 (∀𝑦𝐵 (𝜑𝜓) ↔ (∀𝑦𝐵 𝜑 ∧ ∀𝑦𝐵 𝜓))
21ralbii 2384 . 2 (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ ∀𝑥𝐴 (∀𝑦𝐵 𝜑 ∧ ∀𝑦𝐵 𝜓))
3 r19.26 2497 . 2 (∀𝑥𝐴 (∀𝑦𝐵 𝜑 ∧ ∀𝑦𝐵 𝜓) ↔ (∀𝑥𝐴𝑦𝐵 𝜑 ∧ ∀𝑥𝐴𝑦𝐵 𝜓))
42, 3bitri 182 1 (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴𝑦𝐵 𝜑 ∧ ∀𝑥𝐴𝑦𝐵 𝜓))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  wral 2359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-4 1445  ax-17 1464
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-ral 2364
This theorem is referenced by:  fununi  5082
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