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Theorem r19.26-2 3094
 Description: Restricted quantifier version of 19.26-2 1839. Version of r19.26 3093 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 3093 . . 3 (∀𝑦𝐵 (𝜑𝜓) ↔ (∀𝑦𝐵 𝜑 ∧ ∀𝑦𝐵 𝜓))
21ralbii 3009 . 2 (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ ∀𝑥𝐴 (∀𝑦𝐵 𝜑 ∧ ∀𝑦𝐵 𝜓))
3 r19.26 3093 . 2 (∀𝑥𝐴 (∀𝑦𝐵 𝜑 ∧ ∀𝑦𝐵 𝜓) ↔ (∀𝑥𝐴𝑦𝐵 𝜑 ∧ ∀𝑥𝐴𝑦𝐵 𝜓))
42, 3bitri 264 1 (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴𝑦𝐵 𝜑 ∧ ∀𝑥𝐴𝑦𝐵 𝜓))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 383  ∀wral 2941 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777 This theorem depends on definitions:  df-bi 197  df-an 385  df-ral 2946 This theorem is referenced by:  fununi  6002  tz7.48lem  7581  isffth2  16623  ispos2  16995  issgrpv  17333  issgrpn0  17334  isnsg2  17671  efgred  18207  dfrhm2  18765  cpmatacl  20569  cpmatmcllem  20571  caucfil  23127  aalioulem6  24137  ajmoi  27842  adjmo  28819  iccllysconn  31358  dfso3  31727  ispridl2  33967  ishlat2  34958  fiinfi  38195  ntrk1k3eqk13  38665  isrnghm  42217
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