Theorem nfcsb 3106

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 1476	. . 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 3104	. 2 |- ( T. -> F/_ x [_ A / y ]_ B )
7	6	mptru 1372	1 |- F/_ x [_ A / y ]_ B

Syntax hints:   T. wtru 1364   F/_wnfc 2316   [_csb 3069

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-sbc 2975  df-csb 3070

This theorem is referenced by:  cbvralcsf  3131  cbvrexcsf  3132  cbvreucsf  3133  cbvrabcsf  3134  elfvmptrab1  5623  fmptcof  5696  mpomptsx  6212  dmmpossx  6214  fmpox  6215  fmpoco  6231  dfmpo  6238  f1od2  6250  nfsum  11379  fsum2dlemstep  11456  fisumcom2  11460  nfcprod  11577  fsumcncntop  14352