Theorem nfsbcdw 3760

Description: Deduction version of nfsbcw 3761. Version of nfsbcd 3763 with a disjoint variable condition, which does not require ax-13 2402. (Contributed by NM, 23-Nov-2005.) Avoid ax-13 2402. (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 3740	. 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 2944	. . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})
6	2, 5	nfeld 2934	. 2 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑦 ∣ 𝜓})
7	1, 6	nfxfrd 1887	1 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)

This proof depends on syntax axioms:   → wi 4  Ⅎwnf 1816   ∈ wcel 2145  {cab 2739  Ⅎwnfc 2908  [wsbc 3739

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-sbc 3740

This theorem is used by:  nfsbcw  3761  nfcsbw  3873  sbcnestgfw  4379