Theorem nfsbcdw 3065

Description: Version of nfsbcd 2956 with a disjoint variable condition. (Contributed by NM, 23-Nov-2005.) (Revised by Gino Giotto, 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 2938	. 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 2318	. . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})
6	2, 5	nfeld 2315	. 2 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑦 ∣ 𝜓})
7	1, 6	nfxfrd 1455	1 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)

Syntax hints:   → wi 4  Ⅎwnf 1440   ∈ wcel 2128  {cab 2143  Ⅎwnfc 2286  [wsbc 2937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-sbc 2938

This theorem is referenced by:  nfcsbw  3067