Theorem nfsbc1d 3765

Description: Deduction version of nfsbc1 3766. (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 3748	. 2 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝐴 ∈ {𝑥 ∣ 𝜓})
2		nfsbc1d.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴)
3		nfab1 2930	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜓}
4	3	a1i 11	. . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑥 ∣ 𝜓})
5	2, 4	nfeld 2939	. 2 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑥 ∣ 𝜓})
6	1, 5	nfxfrd 1887	1 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝜓)

This proof depends on syntax axioms:   → wi 4  Ⅎwnf 1816   ∈ wcel 2146  {cab 2744  Ⅎwnfc 2913  [wsbc 3747

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-sbc 3748

This theorem is used by:  nfsbc1  3766  nfcsb1d  3878