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Theorem r19.26-3 2499
Description: Theorem 19.26 of [Margaris] p. 90 with 3 restricted 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 926 . . 3 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
21ralbii 2384 . 2 (∀𝑥𝐴 (𝜑𝜓𝜒) ↔ ∀𝑥𝐴 ((𝜑𝜓) ∧ 𝜒))
3 r19.26 2497 . 2 (∀𝑥𝐴 ((𝜑𝜓) ∧ 𝜒) ↔ (∀𝑥𝐴 (𝜑𝜓) ∧ ∀𝑥𝐴 𝜒))
4 r19.26 2497 . . . 4 (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓))
54anbi1i 446 . . 3 ((∀𝑥𝐴 (𝜑𝜓) ∧ ∀𝑥𝐴 𝜒) ↔ ((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓) ∧ ∀𝑥𝐴 𝜒))
6 df-3an 926 . . 3 ((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓 ∧ ∀𝑥𝐴 𝜒) ↔ ((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓) ∧ ∀𝑥𝐴 𝜒))
75, 6bitr4i 185 . 2 ((∀𝑥𝐴 (𝜑𝜓) ∧ ∀𝑥𝐴 𝜒) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓 ∧ ∀𝑥𝐴 𝜒))
82, 3, 73bitri 204 1 (∀𝑥𝐴 (𝜑𝜓𝜒) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓 ∧ ∀𝑥𝐴 𝜒))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  w3a 924  wral 2359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-4 1445  ax-17 1464
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-ral 2364
This theorem is referenced by: (None)
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