Theorem nfsbc 2983

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 1466	. . 3 ⊢ Ⅎ𝑦⊤
2		nfsbc.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	a1i 9	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐴)
4		nfsbc.2	. . . 4 ⊢ Ⅎ𝑥𝜑
5	4	a1i 9	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
6	1, 3, 5	nfsbcd 2982	. 2 ⊢ (⊤ → Ⅎ𝑥[𝐴 / 𝑦]𝜑)
7	6	mptru 1362	1 ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑

Syntax hints:  ⊤wtru 1354  Ⅎwnf 1460  Ⅎwnfc 2306  [wsbc 2962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-sbc 2963

This theorem is referenced by:  cbvralcsf  3119  cbvrexcsf  3120  opelopabf  4271  ralrnmpt  5654  rexrnmpt  5655  dfopab2  6184  dfoprab3s  6185  mpoxopoveq  6235