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Theorem nfsbc 2757
Description: Bound-variable hypothesis builder for class substitution. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.)
Hypotheses
Ref Expression
nfsbc.1 xA
nfsbc.2 xφ
Assertion
Ref Expression
nfsbc x[A / y]φ

Proof of Theorem nfsbc
StepHypRef Expression
1 nftru 1331 . . 3 y
2 nfsbc.1 . . . 4 xA
32a1i 9 . . 3 ( ⊤ → xA)
4 nfsbc.2 . . . 4 xφ
54a1i 9 . . 3 ( ⊤ → Ⅎxφ)
61, 3, 5nfsbcd 2756 . 2 ( ⊤ → Ⅎx[A / y]φ)
76trud 1235 1 x[A / y]φ
Colors of variables: wff set class
Syntax hints:  wtru 1227  wnf 1325  wnfc 2143  [wsbc 2737
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-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-sbc 2738
This theorem is referenced by:  cbvralcsf  2881  cbvrexcsf  2882  opelopabf  3981  ralrnmpt  5230  rexrnmpt  5231  dfopab2  5734  dfoprab3s  5735  mpt2xopoveq  5773
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