Theorem nfcsb 3175

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 3173	. 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 3137

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 3042  df-csb 3138

This theorem is referenced by:  cbvralcsf  3200  cbvrexcsf  3201  cbvreucsf  3202  cbvrabcsf  3203  elfvmptrab1  5771  fmptcof  5843  mpomptsx  6392  dmmpossx  6394  fmpox  6395  fmpoco  6411  dfmpo  6418  f1od2  6430  nfsum  12038  fsum2dlemstep  12116  fisumcom2  12120  nfcprod  12237  fsumcncntop  15424