Theorem nfcsbd 3170

Description: Deduction version of nfcsb 3171. (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 3138	. 2 ⊢ [A / y]B = {z ∣ [̣A / y]̣z ∈ B}
2		nfv 1619	. . 3 ⊢ Ⅎzφ
3		nfcsbd.1	. . . 4 ⊢ Ⅎyφ
4		nfcsbd.2	. . . 4 ⊢ (φ → ℲxA)
5		nfcsbd.3	. . . . 5 ⊢ (φ → ℲxB)
6	5	nfcrd 2503	. . . 4 ⊢ (φ → Ⅎx z ∈ B)
7	3, 4, 6	nfsbcd 3067	. . 3 ⊢ (φ → Ⅎx[̣A / y]̣z ∈ B)
8	2, 7	nfabd 2509	. 2 ⊢ (φ → Ⅎx{z ∣ [̣A / y]̣z ∈ B})
9	1, 8	nfcxfrd 2488	1 ⊢ (φ → Ⅎx[A / y]B)

This proof depends on syntax axioms:   → wi 4  Ⅎwnf 1544   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2477  [̣wsbc 3047  [csb 3137

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-sbc 3048  df-csb 3138

This theorem is used by:  nfcsb  3171