Theorem nfsbc1d 3756

Description: Deduction version of nfsbc1 3757. (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 3739	. 2 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝐴 ∈ {𝑥 ∣ 𝜓})
2		nfsbc1d.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴)
3		nfab1 2924	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜓}
4	3	a1i 11	. . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑥 ∣ 𝜓})
5	2, 4	nfeld 2933	. 2 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑥 ∣ 𝜓})
6	1, 5	nfxfrd 1886	1 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝜓)

This proof depends on syntax axioms:   → wi 4  Ⅎwnf 1815   ∈ wcel 2144  {cab 2738  Ⅎwnfc 2907  [wsbc 3738

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-sbc 3739

This theorem is used by:  nfsbc1  3757  nfcsb1d  3868