Theorem nfsbc 2971

Description: Bound-variable hypothesis builder for class substitution. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.)

Hypotheses

Ref	Expression
nfsbc.1	|- F/_ x A
nfsbc.2	|- F/ x ph

Assertion

Ref	Expression
nfsbc	|- F/ x [. A / y ]. ph

Proof of Theorem nfsbc

Step	Hyp	Ref	Expression
1		nftru 1454	. . 3 |- F/ y T.
2		nfsbc.1	. . . 4 |- F/_ x A
3	2	a1i 9	. . 3 |- ( T. -> F/_ x A )
4		nfsbc.2	. . . 4 |- F/ x ph
5	4	a1i 9	. . 3 |- ( T. -> F/ x ph )
6	1, 3, 5	nfsbcd 2970	. 2 |- ( T. -> F/ x [. A / y ]. ph )
7	6	mptru 1352	1 |- F/ x [. A / y ]. ph

Syntax hints:   T. wtru 1344   F/wnf 1448   F/_wnfc 2295   [.wsbc 2951

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-sbc 2952

This theorem is referenced by:  cbvralcsf  3107  cbvrexcsf  3108  opelopabf  4252  ralrnmpt  5627  rexrnmpt  5628  dfopab2  6157  dfoprab3s  6158  mpoxopoveq  6208