Theorem nfsbcw 3751

Description: Bound-variable hypothesis builder for class substitution. Version of nfsbc 3754 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 3750	. 2 ⊢ (⊤ → Ⅎ𝑥[𝐴 / 𝑦]𝜑)
7	6	mptru 1549	1 ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑

Syntax hints:  ⊤wtru 1543  Ⅎwnf 1785  Ⅎwnfc 2884  [wsbc 3729

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 3730

This theorem is referenced by:  opelopabgf  5486  opelopabf  5491  ralrnmptw  7038  elovmporab  7604  elovmporab1w  7605  ovmpt3rabdm  7617  elovmpt3rab1  7618  dfopab2  7996  dfoprab3s  7997  ralxpes  8077  ralxp3es  8080  frpoins3xpg  8081  frpoins3xp3g  8082  mpoxopoveq  8160  elmptrab  23801  bnj1445  35207  bnj1446  35208  bnj1467  35217  indexa  38065  sdclem1  38075  sbcalf  38446  sbcexf  38447  sbccomieg  43236  rexrabdioph  43237  or2expropbilem2  47478  or2expropbi  47479  ich2exprop  47928  ichnreuop  47929  reuopreuprim  47983