Theorem nfsbc1d 3453

Description: Deduction version of nfsbc1 3454. (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 3436	. 2 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝐴 ∈ {𝑥 ∣ 𝜓})
2		nfsbc1d.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴)
3		nfab1 2766	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜓}
4	3	a1i 11	. . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑥 ∣ 𝜓})
5	2, 4	nfeld 2773	. 2 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑥 ∣ 𝜓})
6	1, 5	nfxfrd 1780	1 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝜓)

Syntax hints:   → wi 4  Ⅎwnf 1708   ∈ wcel 1990  {cab 2608  Ⅎwnfc 2751  [wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-sbc 3436

This theorem is referenced by:  nfsbc1  3454  nfcsb1d  3547