Theorem nfsbc 2836

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 1396	. . 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 2835	. 2 |- ( T. -> F/ x [. A / y ]. ph )
7	6	trud 1294	1 |- F/ x [. A / y ]. ph

Syntax hints:   T. wtru 1286   F/wnf 1390   F/_wnfc 2207   [.wsbc 2816

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-sbc 2817

This theorem is referenced by:  cbvralcsf  2965  cbvrexcsf  2966  opelopabf  4031  ralrnmpt  5335  rexrnmpt  5336  dfopab2  5840  dfoprab3s  5841  mpt2xopoveq  5883