Theorem nfcsb 3131

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 1489	. . 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 3129	. 2 |- ( T. -> F/_ x [_ A / y ]_ B )
7	6	mptru 1382	1 |- F/_ x [_ A / y ]_ B

Syntax hints:   T. wtru 1374   F/_wnfc 2335   [_csb 3093

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-sbc 2999  df-csb 3094

This theorem is referenced by:  cbvralcsf  3156  cbvrexcsf  3157  cbvreucsf  3158  cbvrabcsf  3159  elfvmptrab1  5676  fmptcof  5749  mpomptsx  6285  dmmpossx  6287  fmpox  6288  fmpoco  6304  dfmpo  6311  f1od2  6323  nfsum  11701  fsum2dlemstep  11778  fisumcom2  11782  nfcprod  11899  fsumcncntop  15072