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Theorem 19.26-3an 1788
 Description: Theorem 19.26 1786 with triple conjunction. (Contributed by NM, 13-Sep-2011.)
Assertion
Ref Expression
19.26-3an (∀𝑥(𝜑𝜓𝜒) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓 ∧ ∀𝑥𝜒))

Proof of Theorem 19.26-3an
StepHypRef Expression
1 19.26 1786 . . 3 (∀𝑥((𝜑𝜓) ∧ 𝜒) ↔ (∀𝑥(𝜑𝜓) ∧ ∀𝑥𝜒))
2 19.26 1786 . . . 4 (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
32anbi1i 727 . . 3 ((∀𝑥(𝜑𝜓) ∧ ∀𝑥𝜒) ↔ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ∧ ∀𝑥𝜒))
41, 3bitri 263 . 2 (∀𝑥((𝜑𝜓) ∧ 𝜒) ↔ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ∧ ∀𝑥𝜒))
5 df-3an 1033 . . 3 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
65albii 1737 . 2 (∀𝑥(𝜑𝜓𝜒) ↔ ∀𝑥((𝜑𝜓) ∧ 𝜒))
7 df-3an 1033 . 2 ((∀𝑥𝜑 ∧ ∀𝑥𝜓 ∧ ∀𝑥𝜒) ↔ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ∧ ∀𝑥𝜒))
84, 6, 73bitr4i 291 1 (∀𝑥(𝜑𝜓𝜒) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓 ∧ ∀𝑥𝜒))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728 This theorem depends on definitions:  df-bi 196  df-an 385  df-3an 1033 This theorem is referenced by:  alrim3con13v  37764  19.21a3con13vVD  38109
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