Theorem nfcsbd 3910

Description: Deduction version of nfcsb 3912. Usage of this theorem is discouraged because it depends on ax-13 2390. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.)

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 3886	. 2 ⊢ ⦋𝐴 / 𝑦⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵}
2		nfv 1915	. . 3 ⊢ Ⅎ𝑧𝜑
3		nfcsbd.1	. . . 4 ⊢ Ⅎ𝑦𝜑
4		nfcsbd.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5		nfcsbd.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐵)
6	5	nfcrd 2971	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ 𝐵)
7	3, 4, 6	nfsbcd 3798	. . 3 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝑧 ∈ 𝐵)
8	2, 7	nfabd 3003	. 2 ⊢ (𝜑 → Ⅎ𝑥{𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵})
9	1, 8	nfcxfrd 2978	1 ⊢ (𝜑 → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵)

Syntax hints:   → wi 4  Ⅎwnf 1784   ∈ wcel 2114  {cab 2801  Ⅎwnfc 2963  [wsbc 3774  ⦋csb 3885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-13 2390  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-sbc 3775  df-csb 3886

This theorem is referenced by:  nfcsb  3912