Theorem nfsbcd 3740

Description: Deduction version of nfsbc 3741. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker nfsbcdw 3737 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 3717	. 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 2932	. . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})
6	2, 5	nfeld 2918	. 2 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑦 ∣ 𝜓})
7	1, 6	nfxfrd 1856	1 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)

Syntax hints:   → wi 4  Ⅎwnf 1786   ∈ wcel 2106  {cab 2715  Ⅎwnfc 2887  [wsbc 3716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2372  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-sbc 3717

This theorem is referenced by:  nfsbc  3741  nfcsbd  3858  sbcnestgf  4357