Theorem nfsbcw 3778

Description: Bound-variable hypothesis builder for class substitution. Version of nfsbc 3781 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 7-Sep-2014.) Avoid ax-13 2371. (Revised by GG, 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 1804	. . 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 3777	. 2 ⊢ (⊤ → Ⅎ𝑥[𝐴 / 𝑦]𝜑)
7	6	mptru 1547	1 ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑

Syntax hints:  ⊤wtru 1541  Ⅎwnf 1783  Ⅎwnfc 2877  [wsbc 3756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-sbc 3757

This theorem is referenced by:  opelopabgf  5503  opelopabf  5508  ralrnmptw  7069  elovmporab  7638  elovmporab1w  7639  ovmpt3rabdm  7651  elovmpt3rab1  7652  dfopab2  8034  dfoprab3s  8035  ralxpes  8118  ralxp3es  8121  frpoins3xpg  8122  frpoins3xp3g  8123  mpoxopoveq  8201  elmptrab  23721  bnj1445  35041  bnj1446  35042  bnj1467  35051  indexa  37734  sdclem1  37744  sbcalf  38115  sbcexf  38116  sbccomieg  42788  rexrabdioph  42789  or2expropbilem2  47038  or2expropbi  47039  ich2exprop  47476  ichnreuop  47477  reuopreuprim  47531