Theorem nfsbc1d 3760

Description: Deduction version of nfsbc1 3761. (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 3743	. 2 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝐴 ∈ {𝑥 ∣ 𝜓})
2		nfsbc1d.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴)
3		nfab1 2901	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜓}
4	3	a1i 11	. . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑥 ∣ 𝜓})
5	2, 4	nfeld 2911	. 2 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑥 ∣ 𝜓})
6	1, 5	nfxfrd 1856	1 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝜓)

Syntax hints:   → wi 4  Ⅎwnf 1785   ∈ wcel 2114  {cab 2715  Ⅎwnfc 2884  [wsbc 3742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-sbc 3743

This theorem is referenced by:  nfsbc1  3761  nfcsb1d  3873