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Theorem r19.26-2 3121
Description: Restricted quantifier version of 19.26-2 1834. Version of r19.26 3120 with two 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 3120 . . 3 (∀𝑦𝐵 (𝜑𝜓) ↔ (∀𝑦𝐵 𝜑 ∧ ∀𝑦𝐵 𝜓))
21ralbii 3115 . 2 (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ ∀𝑥𝐴 (∀𝑦𝐵 𝜑 ∧ ∀𝑦𝐵 𝜓))
3 r19.26 3120 . 2 (∀𝑥𝐴 (∀𝑦𝐵 𝜑 ∧ ∀𝑦𝐵 𝜓) ↔ (∀𝑥𝐴𝑦𝐵 𝜑 ∧ ∀𝑥𝐴𝑦𝐵 𝜓))
42, 3bitri 267 1 (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴𝑦𝐵 𝜑 ∧ ∀𝑥𝐴𝑦𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 387  wral 3088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772
This theorem depends on definitions:  df-bi 199  df-an 388  df-ral 3093
This theorem is referenced by:  fununi  6262  tz7.48lem  7880  isffth2  17044  ispos2  17416  issgrpv  17754  issgrpn0  17755  isnsg2  18093  efgred  18634  dfrhm2  19192  cpmatacl  21028  cpmatmcllem  21030  caucfil  23589  aalioulem6  24629  ajmoi  28413  adjmo  29390  iccllysconn  32088  dfso3  32476  fvineqsnf1  34138  ispridl2  34764  ishlat2  35940  fiinfi  39300  ntrk1k3eqk13  39769  isrnghm  43533
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