Theorem nfcsbd 3075

Description: Deduction version of nfcsb 3077. (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 3041	. 2 ⊢ ⦋𝐴 / 𝑦⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵}
2		nfv 1515	. . 3 ⊢ Ⅎ𝑧𝜑
3		nfcsbd.1	. . . 4 ⊢ Ⅎ𝑦𝜑
4		nfcsbd.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5		nfcsbd.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐵)
6	5	nfcrd 2320	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ 𝐵)
7	3, 4, 6	nfsbcd 2965	. . 3 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝑧 ∈ 𝐵)
8	2, 7	nfabd 2326	. 2 ⊢ (𝜑 → Ⅎ𝑥{𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵})
9	1, 8	nfcxfrd 2304	1 ⊢ (𝜑 → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵)

Syntax hints:   → wi 4  Ⅎwnf 1447   ∈ wcel 2135  {cab 2150  Ⅎwnfc 2293  [wsbc 2946  ⦋csb 3040

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-sbc 2947  df-csb 3041

This theorem is referenced by:  nfcsb  3077