Theorem nfcsb 3176

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 1515	. . 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 3174	. 2 |- ( T. -> F/_ x [_ A / y ]_ B )
7	6	mptru 1407	1 |- F/_ x [_ A / y ]_ B

Syntax hints:   T. wtru 1399   F/_wnfc 2371   [_csb 3138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-sbc 3043  df-csb 3139

This theorem is referenced by:  cbvralcsf  3201  cbvrexcsf  3202  cbvreucsf  3203  cbvrabcsf  3204  elfvmptrab1  5772  fmptcof  5844  mpomptsx  6393  dmmpossx  6395  fmpox  6396  fmpoco  6412  dfmpo  6419  f1od2  6431  nfsum  12042  fsum2dlemstep  12120  fisumcom2  12124  nfcprod  12241  fsumcncntop  15432