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Theorem r2alf 2649
 Description: Double restricted universal quantification. (Contributed by Mario Carneiro, 14-Oct-2016.)
Hypothesis
Ref Expression
r2alf.1 yA
Assertion
Ref Expression
r2alf (x A y B φxy((x A y B) → φ))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)   A(x,y)   B(x,y)

Proof of Theorem r2alf
StepHypRef Expression
1 df-ral 2619 . 2 (x A y B φx(x Ay B φ))
2 r2alf.1 . . . . . 6 yA
32nfcri 2483 . . . . 5 y x A
4319.21 1796 . . . 4 (y(x A → (y Bφ)) ↔ (x Ay(y Bφ)))
5 impexp 433 . . . . 5 (((x A y B) → φ) ↔ (x A → (y Bφ)))
65albii 1566 . . . 4 (y((x A y B) → φ) ↔ y(x A → (y Bφ)))
7 df-ral 2619 . . . . 5 (y B φy(y Bφ))
87imbi2i 303 . . . 4 ((x Ay B φ) ↔ (x Ay(y Bφ)))
94, 6, 83bitr4i 268 . . 3 (y((x A y B) → φ) ↔ (x Ay B φ))
109albii 1566 . 2 (xy((x A y B) → φ) ↔ x(x Ay B φ))
111, 10bitr4i 243 1 (x A y B φxy((x A y B) → φ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   ∈ wcel 1710  Ⅎwnfc 2476  ∀wral 2614 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-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:  r2al  2651  ralcomf  2769
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