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Theorem rgen3 2711
 Description: Generalization rule for restricted quantification. (Contributed by NM, 12-Jan-2008.)
Hypothesis
Ref Expression
rgen3.1 ((x A y B z C) → φ)
Assertion
Ref Expression
rgen3 x A y B z C φ
Distinct variable groups:   y,z,A   z,B   x,y,z
Allowed substitution hints:   φ(x,y,z)   A(x)   B(x,y)   C(x,y,z)

Proof of Theorem rgen3
StepHypRef Expression
1 rgen3.1 . . . 4 ((x A y B z C) → φ)
213expa 1151 . . 3 (((x A y B) z C) → φ)
32ralrimiva 2697 . 2 ((x A y B) → z C φ)
43rgen2 2710 1 x A y B z C φ
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934   ∈ wcel 1710  ∀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-11 1746 This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1542  df-nf 1545  df-ral 2619 This theorem is referenced by: (None)
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