Theorem nfcsb 3166

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 3164	. 2 |- ( T. -> F/_ x [_ A / y ]_ B )
7	6	mptru 1407	1 |- F/_ x [_ A / y ]_ B

Syntax hints:   T. wtru 1399   F/_wnfc 2362   [_csb 3128

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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-sbc 3033  df-csb 3129

This theorem is referenced by:  cbvralcsf  3191  cbvrexcsf  3192  cbvreucsf  3193  cbvrabcsf  3194  elfvmptrab1  5750  fmptcof  5822  mpomptsx  6371  dmmpossx  6373  fmpox  6374  fmpoco  6390  dfmpo  6397  f1od2  6409  nfsum  11997  fsum2dlemstep  12075  fisumcom2  12079  nfcprod  12196  fsumcncntop  15378