Theorem nfsbcd 3802

Description: Deduction version of nfsbc 3803. Usage of this theorem is discouraged because it depends on ax-13 2366. Use the weaker nfsbcdw 3799 when possible. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfsbcd.1	⊢ Ⅎ𝑦𝜑
nfsbcd.2	⊢ (𝜑 → Ⅎ𝑥𝐴)
nfsbcd.3	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfsbcd	⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)

Proof of Theorem nfsbcd

Step	Hyp	Ref	Expression
1		df-sbc 3779	. 2 ⊢ ([𝐴 / 𝑦]𝜓 ↔ 𝐴 ∈ {𝑦 ∣ 𝜓})
2		nfsbcd.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴)
3		nfsbcd.1	. . . 4 ⊢ Ⅎ𝑦𝜑
4		nfsbcd.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
5	3, 4	nfabd 2925	. . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})
6	2, 5	nfeld 2911	. 2 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑦 ∣ 𝜓})
7	1, 6	nfxfrd 1848	1 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)

Syntax hints:   → wi 4  Ⅎwnf 1777   ∈ wcel 2098  {cab 2705  Ⅎwnfc 2879  [wsbc 3778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-13 2366  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-sbc 3779

This theorem is referenced by:  nfsbc  3803  nfcsbd  3920  sbcnestgf  4427