Theorem nfsbcdw 3774

Description: Deduction version of nfsbcw 3775. Version of nfsbcd 3777 with a disjoint variable condition, which does not require ax-13 2410. (Contributed by NM, 23-Nov-2005.) Avoid ax-13 2410. (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 3754	. 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 2952	. . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})
6	2, 5	nfeld 2942	. 2 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑦 ∣ 𝜓})
7	1, 6	nfxfrd 1881	1 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)

Syntax hints:   → wi 4  Ⅎwnf 1810   ∈ wcel 2149  {cab 2747  Ⅎwnfc 2916  [wsbc 3753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-sbc 3754

This theorem is referenced by:  nfsbcw  3775  nfcsbw  3887  sbcnestgfw  4384