Theorem nfcsb 3068

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 1446	. . 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 3066	. 2 |- ( T. -> F/_ x [_ A / y ]_ B )
7	6	mptru 1344	1 |- F/_ x [_ A / y ]_ B

Syntax hints:   T. wtru 1336   F/_wnfc 2286   [_csb 3031

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-sbc 2938  df-csb 3032

This theorem is referenced by:  cbvralcsf  3093  cbvrexcsf  3094  cbvreucsf  3095  cbvrabcsf  3096  elfvmptrab1  5562  fmptcof  5634  mpomptsx  6145  dmmpossx  6147  fmpox  6148  fmpoco  6163  dfmpo  6170  f1od2  6182  nfsum  11254  fsum2dlemstep  11331  fisumcom2  11335  nfcprod  11452  fsumcncntop  12967