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Theorem r19.26-3 2443
 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 887 . . 3 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
21ralbii 2330 . 2 (∀𝑥𝐴 (𝜑𝜓𝜒) ↔ ∀𝑥𝐴 ((𝜑𝜓) ∧ 𝜒))
3 r19.26 2441 . 2 (∀𝑥𝐴 ((𝜑𝜓) ∧ 𝜒) ↔ (∀𝑥𝐴 (𝜑𝜓) ∧ ∀𝑥𝐴 𝜒))
4 r19.26 2441 . . . 4 (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓))
54anbi1i 431 . . 3 ((∀𝑥𝐴 (𝜑𝜓) ∧ ∀𝑥𝐴 𝜒) ↔ ((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓) ∧ ∀𝑥𝐴 𝜒))
6 df-3an 887 . . 3 ((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓 ∧ ∀𝑥𝐴 𝜒) ↔ ((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓) ∧ ∀𝑥𝐴 𝜒))
75, 6bitr4i 176 . 2 ((∀𝑥𝐴 (𝜑𝜓) ∧ ∀𝑥𝐴 𝜒) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓 ∧ ∀𝑥𝐴 𝜒))
82, 3, 73bitri 195 1 (∀𝑥𝐴 (𝜑𝜓𝜒) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓 ∧ ∀𝑥𝐴 𝜒))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   ∧ w3a 885  ∀wral 2306 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-4 1400  ax-17 1419 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-ral 2311 This theorem is referenced by: (None)
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