Theorem nfcsb 3185

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 1519	. . 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 3183	. 2 |- ( T. -> F/_ x [_ A / y ]_ B )
7	6	mptru 1411	1 |- F/_ x [_ A / y ]_ B

Syntax hints:   T. wtru 1403   F/_wnfc 2379   [_csb 3147

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-sbc 3052  df-csb 3148

This theorem is referenced by:  cbvralcsf  3210  cbvrexcsf  3211  cbvreucsf  3212  cbvrabcsf  3213  elfvmptrab1  5794  fmptcof  5866  mpomptsx  6423  dmmpossx  6425  fmpox  6426  fmpoco  6442  dfmpo  6449  f1od2  6461  nfsum  12101  fsum2dlemstep  12179  fisumcom2  12183  nfcprod  12300  fsumcncntop  15591