Theorem nfsbc1d 3064

Description: Deduction version of nfsbc1 3065. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)

Hypothesis

Ref	Expression
nfsbc1d.2	⊢ (φ → ℲxA)

Assertion

Ref	Expression
nfsbc1d	⊢ (φ → Ⅎx[̣A / x]̣ψ)

Proof of Theorem nfsbc1d

Step	Hyp	Ref	Expression
1		df-sbc 3048	. 2 ⊢ ([̣A / x]̣ψ ↔ A ∈ {x ∣ ψ})
2		nfsbc1d.2	. . 3 ⊢ (φ → ℲxA)
3		nfab1 2492	. . . 4 ⊢ Ⅎx{x ∣ ψ}
4	3	a1i 10	. . 3 ⊢ (φ → Ⅎx{x ∣ ψ})
5	2, 4	nfeld 2505	. 2 ⊢ (φ → Ⅎx A ∈ {x ∣ ψ})
6	1, 5	nfxfrd 1571	1 ⊢ (φ → Ⅎx[̣A / x]̣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:  nfsbc1  3065  nfcsb1d  3167