Theorem nfcsbd 3080

Description: Deduction version of nfcsb 3082. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)

Hypotheses

Ref	Expression
nfcsbd.1	⊢ Ⅎ𝑦𝜑
nfcsbd.2	⊢ (𝜑 → Ⅎ𝑥𝐴)
nfcsbd.3	⊢ (𝜑 → Ⅎ𝑥𝐵)

Assertion

Ref	Expression
nfcsbd	⊢ (𝜑 → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵)

Proof of Theorem nfcsbd

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 3046	. 2 ⊢ ⦋𝐴 / 𝑦⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵}
2		nfv 1516	. . 3 ⊢ Ⅎ𝑧𝜑
3		nfcsbd.1	. . . 4 ⊢ Ⅎ𝑦𝜑
4		nfcsbd.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5		nfcsbd.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐵)
6	5	nfcrd 2322	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ 𝐵)
7	3, 4, 6	nfsbcd 2970	. . 3 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝑧 ∈ 𝐵)
8	2, 7	nfabd 2328	. 2 ⊢ (𝜑 → Ⅎ𝑥{𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵})
9	1, 8	nfcxfrd 2306	1 ⊢ (𝜑 → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵)

Syntax hints:   → wi 4  Ⅎwnf 1448   ∈ wcel 2136  {cab 2151  Ⅎwnfc 2295  [wsbc 2951  ⦋csb 3045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-sbc 2952  df-csb 3046

This theorem is referenced by:  nfcsb  3082