Theorem nfsbcw 3182

Description: Bound-variable hypothesis builder for class substitution. Version of nfsbc 3072 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 1519	. . 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 3181	. 2 |- ( T. -> F/ x [. A / y ]. ph )
7	6	mptru 1411	1 |- F/ x [. A / y ]. ph

Syntax hints:   T. wtru 1403   F/wnf 1513   F/_wnfc 2379   [.wsbc 3051

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-sbc 3052

This theorem is referenced by:  opelopabgf  4407  elovmporab  6279  elovmporab1w  6280