Theorem nfcsb 3086

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 1459	. . 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 3084	. 2 |- ( T. -> F/_ x [_ A / y ]_ B )
7	6	mptru 1357	1 |- F/_ x [_ A / y ]_ B

Syntax hints:   T. wtru 1349   F/_wnfc 2299   [_csb 3049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-sbc 2956  df-csb 3050

This theorem is referenced by:  cbvralcsf  3111  cbvrexcsf  3112  cbvreucsf  3113  cbvrabcsf  3114  elfvmptrab1  5590  fmptcof  5663  mpomptsx  6176  dmmpossx  6178  fmpox  6179  fmpoco  6195  dfmpo  6202  f1od2  6214  nfsum  11320  fsum2dlemstep  11397  fisumcom2  11401  nfcprod  11518  fsumcncntop  13350