Theorem nfsbcw 3799

Description: Bound-variable hypothesis builder for class substitution. Version of nfsbc 3802 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by NM, 7-Sep-2014.) Avoid ax-13 2370. (Revised by Gino Giotto, 10-Jan-2024.)

Hypotheses

Ref	Expression
nfsbcw.1	⊢ Ⅎ𝑥𝐴
nfsbcw.2	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfsbcw	⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem nfsbcw

Step	Hyp	Ref	Expression
1		nftru 1805	. . 3 ⊢ Ⅎ𝑦⊤
2		nfsbcw.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐴)
4		nfsbcw.2	. . . 4 ⊢ Ⅎ𝑥𝜑
5	4	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
6	1, 3, 5	nfsbcdw 3798	. 2 ⊢ (⊤ → Ⅎ𝑥[𝐴 / 𝑦]𝜑)
7	6	mptru 1547	1 ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑

Syntax hints:  ⊤wtru 1541  Ⅎwnf 1784  Ⅎwnfc 2882  [wsbc 3777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-sbc 3778

This theorem is referenced by:  opelopabgf  5540  opelopabf  5545  ralrnmptw  7095  elovmporab  7656  elovmporab1w  7657  ovmpt3rabdm  7669  elovmpt3rab1  7670  dfopab2  8042  dfoprab3s  8043  ralxpes  8127  ralxp3es  8130  frpoins3xpg  8131  frpoins3xp3g  8132  mpoxopoveq  8210  elmptrab  23650  bnj1445  34518  bnj1446  34519  bnj1467  34528  indexa  37064  sdclem1  37074  sbcalf  37445  sbcexf  37446  sbccomieg  41993  rexrabdioph  41994  or2expropbilem2  46201  or2expropbi  46202  ich2exprop  46597  ichnreuop  46598  reuopreuprim  46652