Theorem nfsbc 3424

Description: Bound-variable hypothesis builder for class substitution. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.)

Hypotheses

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

Assertion

Ref	Expression
nfsbc	⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑

Proof of Theorem nfsbc

Step	Hyp	Ref	Expression
1		nftru 1721	. . 3 ⊢ Ⅎ𝑦⊤
2		nfsbc.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐴)
4		nfsbc.2	. . . 4 ⊢ Ⅎ𝑥𝜑
5	4	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
6	1, 3, 5	nfsbcd 3423	. 2 ⊢ (⊤ → Ⅎ𝑥[𝐴 / 𝑦]𝜑)
7	6	trud 1484	1 ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑

Syntax hints:  ⊤wtru 1476  Ⅎwnf 1699  Ⅎwnfc 2738  [wsbc 3402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-sbc 3403

This theorem is referenced by:  cbvralcsf  3531  opelopabgf  4920  opelopabf  4925  ralrnmpt  6276  elovmpt2rab  6778  elovmpt2rab1  6779  ovmpt3rabdm  6790  elovmpt3rab1  6791  dfopab2  7113  dfoprab3s  7114  mpt2xopoveq  7232  elmptrab  21440  bnj1445  30366  bnj1446  30367  bnj1467  30376  indexa  32698  sdclem1  32709  sbcalf  33087  sbcexf  33088  sbccomieg  36375  rexrabdioph  36376