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Theorem r19.26-2 3132
Description: Restricted quantifier version of 19.26-2 1866. Version of r19.26 3105 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 3105 . . 3 (∀𝑦𝐵 (𝜑𝜓) ↔ (∀𝑦𝐵 𝜑 ∧ ∀𝑦𝐵 𝜓))
21ralbii 3087 . 2 (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ ∀𝑥𝐴 (∀𝑦𝐵 𝜑 ∧ ∀𝑦𝐵 𝜓))
3 r19.26 3105 . 2 (∀𝑥𝐴 (∀𝑦𝐵 𝜑 ∧ ∀𝑦𝐵 𝜓) ↔ (∀𝑥𝐴𝑦𝐵 𝜑 ∧ ∀𝑥𝐴𝑦𝐵 𝜓))
42, 3bitri 275 1 (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴𝑦𝐵 𝜑 ∧ ∀𝑥𝐴𝑦𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  wral 3055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803
This theorem depends on definitions:  df-bi 206  df-an 396  df-ral 3056
This theorem is referenced by:  fununi  6617  tz7.48lem  8442  isffth2  17878  ispos2  18280  issgrpv  18654  issgrpn0  18655  isnsg2  19083  efgred  19668  isrnghm  20343  dfrhm2  20376  df2idl2rng  21113  cpmatacl  22573  cpmatmcllem  22575  caucfil  25166  aalioulem6  26227  ajmoi  30620  adjmo  31594  prmidl2  33065  iccllysconn  34769  dfso3  35223  fvineqsnf1  36798  ispridl2  37419  ishlat2  38736  fiinfi  42900  ntrk1k3eqk13  43377
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