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Theorem r19.26-3 3213
Description: Version of r19.26 3211 with three quantifiers. (Contributed by FL, 22-Nov-2010.)
Assertion
Ref Expression
r19.26-3 (∀𝑥𝐴 (𝜑𝜓𝜒) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓 ∧ ∀𝑥𝐴 𝜒))

Proof of Theorem r19.26-3
StepHypRef Expression
1 df-3an 1072 . . 3 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
21ralbii 3128 . 2 (∀𝑥𝐴 (𝜑𝜓𝜒) ↔ ∀𝑥𝐴 ((𝜑𝜓) ∧ 𝜒))
3 r19.26 3211 . 2 (∀𝑥𝐴 ((𝜑𝜓) ∧ 𝜒) ↔ (∀𝑥𝐴 (𝜑𝜓) ∧ ∀𝑥𝐴 𝜒))
4 r19.26 3211 . . . 4 (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓))
54anbi1i 602 . . 3 ((∀𝑥𝐴 (𝜑𝜓) ∧ ∀𝑥𝐴 𝜒) ↔ ((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓) ∧ ∀𝑥𝐴 𝜒))
6 df-3an 1072 . . 3 ((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓 ∧ ∀𝑥𝐴 𝜒) ↔ ((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓) ∧ ∀𝑥𝐴 𝜒))
75, 6bitr4i 267 . 2 ((∀𝑥𝐴 (𝜑𝜓) ∧ ∀𝑥𝐴 𝜒) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓 ∧ ∀𝑥𝐴 𝜒))
82, 3, 73bitri 286 1 (∀𝑥𝐴 (𝜑𝜓𝜒) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓 ∧ ∀𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  w3a 1070  wral 3060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884
This theorem depends on definitions:  df-bi 197  df-an 383  df-3an 1072  df-ral 3065
This theorem is referenced by:  sgrp2rid2ex  17621  axeuclid  26063  axcontlem8  26071  stoweidlem60  40788
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