Theorem nfsbc 2984

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 1466	. . 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 2983	. 2 |- ( T. -> F/ x [. A / y ]. ph )
7	6	mptru 1362	1 |- F/ x [. A / y ]. ph

Syntax hints:   T. wtru 1354   F/wnf 1460   F/_wnfc 2306   [.wsbc 2963

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-sbc 2964

This theorem is referenced by:  cbvralcsf  3120  cbvrexcsf  3121  opelopabf  4275  ralrnmpt  5659  rexrnmpt  5660  dfopab2  6190  dfoprab3s  6191  mpoxopoveq  6241