Theorem nfsbcd 3067

Description: Deduction version of nfsbc 3068. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)

Hypotheses

Ref	Expression
nfsbcd.1	⊢ Ⅎyφ
nfsbcd.2	⊢ (φ → ℲxA)
nfsbcd.3	⊢ (φ → Ⅎxψ)

Assertion

Ref	Expression
nfsbcd	⊢ (φ → Ⅎx[̣A / y]̣ψ)

Proof of Theorem nfsbcd

Step	Hyp	Ref	Expression
1		df-sbc 3048	. 2 ⊢ ([̣A / y]̣ψ ↔ A ∈ {y ∣ ψ})
2		nfsbcd.2	. . 3 ⊢ (φ → ℲxA)
3		nfsbcd.1	. . . 4 ⊢ Ⅎyφ
4		nfsbcd.3	. . . 4 ⊢ (φ → Ⅎxψ)
5	3, 4	nfabd 2509	. . 3 ⊢ (φ → Ⅎx{y ∣ ψ})
6	2, 5	nfeld 2505	. 2 ⊢ (φ → Ⅎx A ∈ {y ∣ ψ})
7	1, 6	nfxfrd 1571	1 ⊢ (φ → Ⅎx[̣A / y]̣ψ)

Syntax hints:   → wi 4  Ⅎwnf 1544   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2477  [̣wsbc 3047

This theorem was proved from 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 theorem 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

This theorem is referenced by:  nfsbc  3068  nfcsbd  3170  sbcnestgf  3184