Theorem nfsbc1d 3792

Description: Deduction version of nfsbc1 3793. (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 3775	. 2 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝐴 ∈ {𝑥 ∣ 𝜓})
2		nfsbc1d.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴)
3		nfab1 2981	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜓}
4	3	a1i 11	. . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑥 ∣ 𝜓})
5	2, 4	nfeld 2991	. 2 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑥 ∣ 𝜓})
6	1, 5	nfxfrd 1854	1 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝜓)

Syntax hints:   → wi 4  Ⅎwnf 1784   ∈ wcel 2114  {cab 2801  Ⅎwnfc 2963  [wsbc 3774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-sbc 3775

This theorem is referenced by:  nfsbc1  3793  nfcsb1d  3907