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Theorem hbral 2322
 Description: Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by David Abernethy, 13-Dec-2009.)
Hypotheses
Ref Expression
hbral.1 (y Ax y A)
hbral.2 (φxφ)
Assertion
Ref Expression
hbral (y A φxy A φ)

Proof of Theorem hbral
StepHypRef Expression
1 df-ral 2280 . 2 (y A φy(y Aφ))
2 hbral.1 . . . 4 (y Ax y A)
3 hbral.2 . . . 4 (φxφ)
42, 3hbim 1410 . . 3 ((y Aφ) → x(y Aφ))
54hbal 1339 . 2 (y(y Aφ) → xy(y Aφ))
61, 5hbxfrbi 1334 1 (y A φxy A φ)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1221   ∈ wcel 1366  ∀wral 2275 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 1309  ax-7 1310  ax-gen 1311  ax-4 1373  ax-i5r 1401 This theorem depends on definitions:  df-bi 110  df-ral 2280 This theorem is referenced by: (None)
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