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Theorem r3al 2671
 Description: Triple restricted universal quantification. (Contributed by NM, 19-Nov-1995.)
Assertion
Ref Expression
r3al (x A y B z C φxyz((x A y B z C) → φ))
Distinct variable groups:   x,y,z   y,A,z   z,B
Allowed substitution hints:   φ(x,y,z)   A(x)   B(x,y)   C(x,y,z)

Proof of Theorem r3al
StepHypRef Expression
1 df-ral 2619 . 2 (x A yz((y B z C) → φ) ↔ x(x Ayz((y B z C) → φ)))
2 r2al 2651 . . 3 (y B z C φyz((y B z C) → φ))
32ralbii 2638 . 2 (x A y B z C φx A yz((y B z C) → φ))
4 3anass 938 . . . . . . . . 9 ((x A y B z C) ↔ (x A (y B z C)))
54imbi1i 315 . . . . . . . 8 (((x A y B z C) → φ) ↔ ((x A (y B z C)) → φ))
6 impexp 433 . . . . . . . 8 (((x A (y B z C)) → φ) ↔ (x A → ((y B z C) → φ)))
75, 6bitri 240 . . . . . . 7 (((x A y B z C) → φ) ↔ (x A → ((y B z C) → φ)))
87albii 1566 . . . . . 6 (z((x A y B z C) → φ) ↔ z(x A → ((y B z C) → φ)))
9 19.21v 1890 . . . . . 6 (z(x A → ((y B z C) → φ)) ↔ (x Az((y B z C) → φ)))
108, 9bitri 240 . . . . 5 (z((x A y B z C) → φ) ↔ (x Az((y B z C) → φ)))
1110albii 1566 . . . 4 (yz((x A y B z C) → φ) ↔ y(x Az((y B z C) → φ)))
12 19.21v 1890 . . . 4 (y(x Az((y B z C) → φ)) ↔ (x Ayz((y B z C) → φ)))
1311, 12bitri 240 . . 3 (yz((x A y B z C) → φ) ↔ (x Ayz((y B z C) → φ)))
1413albii 1566 . 2 (xyz((x A y B z C) → φ) ↔ x(x Ayz((y B z C) → φ)))
151, 3, 143bitr4i 268 1 (x A y B z C φxyz((x A y B z C) → φ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540   ∈ wcel 1710  ∀wral 2614 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619 This theorem is referenced by: (None)
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