Theorem nfsbcw 3115

Description: Bound-variable hypothesis builder for class substitution. Version of nfsbc 3006 with a disjoint variable condition, which in the future may make it possible to reduce axiom usage. (Contributed by NM, 7-Sep-2014.) (Revised by GG, 10-Jan-2024.)

Hypotheses

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

Assertion

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

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   A( x, y)

Proof of Theorem nfsbcw

Step	Hyp	Ref	Expression
1		nftru 1477	. . 3 |- F/ y T.
2		nfsbcw.1	. . . 4 |- F/_ x A
3	2	a1i 9	. . 3 |- ( T. -> F/_ x A )
4		nfsbcw.2	. . . 4 |- F/ x ph
5	4	a1i 9	. . 3 |- ( T. -> F/ x ph )
6	1, 3, 5	nfsbcdw 3114	. 2 |- ( T. -> F/ x [. A / y ]. ph )
7	6	mptru 1373	1 |- F/ x [. A / y ]. ph

Syntax hints:   T. wtru 1365   F/wnf 1471   F/_wnfc 2323   [.wsbc 2985

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-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-sbc 2986

This theorem is referenced by:  elovmporab  6110  elovmporab1w  6111