Theorem nfcsbd 3116

Description: Deduction version of nfcsb 3118. (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 3081	. 2 ⊢ ⦋𝐴 / 𝑦⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵}
2		nfv 1539	. . 3 ⊢ Ⅎ𝑧𝜑
3		nfcsbd.1	. . . 4 ⊢ Ⅎ𝑦𝜑
4		nfcsbd.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5		nfcsbd.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐵)
6	5	nfcrd 2350	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ 𝐵)
7	3, 4, 6	nfsbcd 3005	. . 3 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝑧 ∈ 𝐵)
8	2, 7	nfabd 2356	. 2 ⊢ (𝜑 → Ⅎ𝑥{𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵})
9	1, 8	nfcxfrd 2334	1 ⊢ (𝜑 → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵)

Syntax hints:   → wi 4  Ⅎwnf 1471   ∈ wcel 2164  {cab 2179  Ⅎwnfc 2323  [wsbc 2985  ⦋csb 3080

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-sbc 2986  df-csb 3081

This theorem is referenced by:  nfcsb  3118