Theorem csbnestgw 4385

Description: Nest the composition of two substitutions. Version of csbnestg 4390 with a disjoint variable condition, which does not require ax-13 2404. (Contributed by NM, 23-Nov-2005.) Avoid ax-13 2404. (Revised by GG, 26-Jan-2024.)

Assertion

Ref	Expression
csbnestgw	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶)

Distinct variable groups:   𝑥,𝑦   𝑥,𝐶

Allowed substitution hints:   𝐴(𝑥, 𝑦)   𝐵(𝑥, 𝑦)   𝐶(𝑦)   𝑉(𝑥, 𝑦)

Proof of Theorem csbnestgw

Step	Hyp	Ref	Expression
1		nfcv 2926	. . 3 ⊢ Ⅎ𝑥𝐶
2	1	ax-gen 1828	. 2 ⊢ ∀𝑦Ⅎ𝑥𝐶
3		csbnestgfw 4383	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝐶) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶)
4	2, 3	mpan2 704	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568   = wceq 1570   ∈ wcel 2147  Ⅎwnfc 2911  ⦋csb 3851

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-v 3456  df-sbc 3743  df-csb 3852

This theorem is used by:  disjxpin  33280  poimirlem24  38830  cdleme31snd  41709  cdlemeg46c  41836