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Theorem r19.26-2 3135
Description: Restricted quantifier version of 19.26-2 1866. Version of r19.26 3108 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 3108 . . 3 (∀𝑦𝐵 (𝜑𝜓) ↔ (∀𝑦𝐵 𝜑 ∧ ∀𝑦𝐵 𝜓))
21ralbii 3090 . 2 (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ ∀𝑥𝐴 (∀𝑦𝐵 𝜑 ∧ ∀𝑦𝐵 𝜓))
3 r19.26 3108 . 2 (∀𝑥𝐴 (∀𝑦𝐵 𝜑 ∧ ∀𝑦𝐵 𝜓) ↔ (∀𝑥𝐴𝑦𝐵 𝜑 ∧ ∀𝑥𝐴𝑦𝐵 𝜓))
42, 3bitri 274 1 (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴𝑦𝐵 𝜑 ∧ ∀𝑥𝐴𝑦𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394  wral 3058
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 395  df-ral 3059
This theorem is referenced by:  fununi  6633  tz7.48lem  8470  isffth2  17914  ispos2  18316  issgrpv  18690  issgrpn0  18691  isnsg2  19125  efgred  19717  isrnghm  20394  dfrhm2  20427  df2idl2rng  21164  cpmatacl  22646  cpmatmcllem  22648  caucfil  25239  aalioulem6  26300  ajmoi  30696  adjmo  31670  prmidl2  33190  iccllysconn  34901  dfso3  35355  fvineqsnf1  36930  ispridl2  37552  ishlat2  38865  fiinfi  43052  ntrk1k3eqk13  43529
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