Theorem nfsbcdw 3812

Description: Deduction version of nfsbcw 3813. Version of nfsbcd 3815 with a disjoint variable condition, which does not require ax-13 2375. (Contributed by NM, 23-Nov-2005.) Avoid ax-13 2375. (Revised by GG, 10-Jan-2024.)

Hypotheses

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

Assertion

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

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfsbcdw

Step	Hyp	Ref	Expression
1		df-sbc 3792	. 2 ⊢ ([𝐴 / 𝑦]𝜓 ↔ 𝐴 ∈ {𝑦 ∣ 𝜓})
2		nfsbcdw.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴)
3		nfsbcdw.1	. . . 4 ⊢ Ⅎ𝑦𝜑
4		nfsbcdw.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
5	3, 4	nfabdw 2925	. . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})
6	2, 5	nfeld 2915	. 2 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑦 ∣ 𝜓})
7	1, 6	nfxfrd 1851	1 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)

Syntax hints:   → wi 4  Ⅎwnf 1780   ∈ wcel 2106  {cab 2712  Ⅎwnfc 2888  [wsbc 3791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-sbc 3792

This theorem is referenced by:  nfsbcw  3813  nfcsbw  3935  sbcnestgfw  4427