Theorem nfsbcw 3775

Description: Bound-variable hypothesis builder for class substitution. Version of nfsbc 3778 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by NM, 7-Sep-2014.) Avoid ax-13 2370. (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 3774	. 2 ⊢ (⊤ → Ⅎ𝑥[𝐴 / 𝑦]𝜑)
7	6	mptru 1547	1 ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑

Syntax hints:  ⊤wtru 1541  Ⅎwnf 1783  Ⅎwnfc 2876  [wsbc 3753

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 2701

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 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-sbc 3754

This theorem is referenced by:  opelopabgf  5500  opelopabf  5505  ralrnmptw  7066  elovmporab  7635  elovmporab1w  7636  ovmpt3rabdm  7648  elovmpt3rab1  7649  dfopab2  8031  dfoprab3s  8032  ralxpes  8115  ralxp3es  8118  frpoins3xpg  8119  frpoins3xp3g  8120  mpoxopoveq  8198  elmptrab  23714  bnj1445  35034  bnj1446  35035  bnj1467  35044  indexa  37727  sdclem1  37737  sbcalf  38108  sbcexf  38109  sbccomieg  42781  rexrabdioph  42782  or2expropbilem2  47034  or2expropbi  47035  ich2exprop  47472  ichnreuop  47473  reuopreuprim  47527