Theorem nfsbcd 3828

Description: Deduction version of nfsbc 3829. Usage of this theorem is discouraged because it depends on ax-13 2380. Use the weaker nfsbcdw 3825 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 3805	. 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 2934	. . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})
6	2, 5	nfeld 2920	. 2 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑦 ∣ 𝜓})
7	1, 6	nfxfrd 1852	1 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)

Syntax hints:   → wi 4  Ⅎwnf 1781   ∈ wcel 2108  {cab 2717  Ⅎwnfc 2893  [wsbc 3804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-13 2380  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-sbc 3805

This theorem is referenced by:  nfsbc  3829  nfcsbd  3947  sbcnestgf  4449