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Theorem nfral 2667
 Description: Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.)
Hypotheses
Ref Expression
nfral.1 xA
nfral.2 xφ
Assertion
Ref Expression
nfral xy A φ

Proof of Theorem nfral
StepHypRef Expression
1 nftru 1554 . . 3 y
2 nfral.1 . . . 4 xA
32a1i 10 . . 3 ( ⊤ → xA)
4 nfral.2 . . . 4 xφ
54a1i 10 . . 3 ( ⊤ → Ⅎxφ)
61, 3, 5nfrald 2665 . 2 ( ⊤ → Ⅎxy A φ)
76trud 1323 1 xy A φ
 Colors of variables: wff setvar class Syntax hints:   ⊤ wtru 1316  Ⅎwnf 1544  Ⅎwnfc 2476  ∀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-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:  nfra2  2668  nfrex  2669  rspc2  2960  sbcralt  3118  sbcralg  3120  raaan  3657  nfint  3936  nfiin  3996  ralxpf  4827  fun11iun  5305  dff13f  5472  nfiso  5487  mpt2eq123  5661  fmpt2x  5730
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