Theorem nfsbcw 3763

Description: Bound-variable hypothesis builder for class substitution. Version of nfsbc 3766 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by NM, 7-Sep-2014.) Avoid ax-13 2377. (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 1806	. . 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 3762	. 2 ⊢ (⊤ → Ⅎ𝑥[𝐴 / 𝑦]𝜑)
7	6	mptru 1549	1 ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑

Syntax hints:  ⊤wtru 1543  Ⅎwnf 1785  Ⅎwnfc 2884  [wsbc 3741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-sbc 3742

This theorem is referenced by:  opelopabgf  5489  opelopabf  5494  ralrnmptw  7041  elovmporab  7606  elovmporab1w  7607  ovmpt3rabdm  7619  elovmpt3rab1  7620  dfopab2  7998  dfoprab3s  7999  ralxpes  8080  ralxp3es  8083  frpoins3xpg  8084  frpoins3xp3g  8085  mpoxopoveq  8163  elmptrab  23775  bnj1445  35181  bnj1446  35182  bnj1467  35191  indexa  37905  sdclem1  37915  sbcalf  38286  sbcexf  38287  sbccomieg  43071  rexrabdioph  43072  or2expropbilem2  47315  or2expropbi  47316  ich2exprop  47753  ichnreuop  47754  reuopreuprim  47808