Theorem nfcsbd 3164

Description: Deduction version of nfcsb 3166. (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 3129	. 2 ⊢ ⦋𝐴 / 𝑦⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵}
2		nfv 1577	. . 3 ⊢ Ⅎ𝑧𝜑
3		nfcsbd.1	. . . 4 ⊢ Ⅎ𝑦𝜑
4		nfcsbd.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5		nfcsbd.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐵)
6	5	nfcrd 2389	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ 𝐵)
7	3, 4, 6	nfsbcd 3052	. . 3 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝑧 ∈ 𝐵)
8	2, 7	nfabd 2395	. 2 ⊢ (𝜑 → Ⅎ𝑥{𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵})
9	1, 8	nfcxfrd 2373	1 ⊢ (𝜑 → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵)

Syntax hints:   → wi 4  Ⅎwnf 1509   ∈ wcel 2202  {cab 2217  Ⅎwnfc 2362  [wsbc 3032  ⦋csb 3128

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-sbc 3033  df-csb 3129

This theorem is referenced by:  nfcsb  3166