Theorem nfcsb 3094

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 1466	. . 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 3092	. 2 |- ( T. -> F/_ x [_ A / y ]_ B )
7	6	mptru 1362	1 |- F/_ x [_ A / y ]_ B

Syntax hints:   T. wtru 1354   F/_wnfc 2306   [_csb 3057

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-sbc 2963  df-csb 3058

This theorem is referenced by:  cbvralcsf  3119  cbvrexcsf  3120  cbvreucsf  3121  cbvrabcsf  3122  elfvmptrab1  5610  fmptcof  5683  mpomptsx  6197  dmmpossx  6199  fmpox  6200  fmpoco  6216  dfmpo  6223  f1od2  6235  nfsum  11364  fsum2dlemstep  11441  fisumcom2  11445  nfcprod  11562  fsumcncntop  14026