Theorem nfcsb 3032

Description: Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)

Hypotheses

Ref	Expression
nfcsb.1	|- F/_ x A
nfcsb.2	|- F/_ x B

Assertion

Ref	Expression
nfcsb	|- F/_ x [_ A / y ]_ B

Proof of Theorem nfcsb

Step	Hyp	Ref	Expression
1		nftru 1442	. . 3 |- F/ y T.
2		nfcsb.1	. . . 4 |- F/_ x A
3	2	a1i 9	. . 3 |- ( T. -> F/_ x A )
4		nfcsb.2	. . . 4 |- F/_ x B
5	4	a1i 9	. . 3 |- ( T. -> F/_ x B )
6	1, 3, 5	nfcsbd 3031	. 2 |- ( T. -> F/_ x [_ A / y ]_ B )
7	6	mptru 1340	1 |- F/_ x [_ A / y ]_ B

Syntax hints:   T. wtru 1332   F/_wnfc 2266   [_csb 2998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-sbc 2905  df-csb 2999

This theorem is referenced by:  cbvralcsf  3057  cbvrexcsf  3058  cbvreucsf  3059  cbvrabcsf  3060  elfvmptrab1  5508  fmptcof  5580  mpomptsx  6088  dmmpossx  6090  fmpox  6091  fmpoco  6106  dfmpo  6113  f1od2  6125  nfsum  11119  fsum2dlemstep  11196  fisumcom2  11200  nfcprod  11317  fsumcncntop  12714