Theorem csbnestgw 4421

Description: Nest the composition of two substitutions. Version of csbnestg 4426 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 23-Nov-2005.) Avoid ax-13 2371. (Revised by Gino Giotto, 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 2903	. . 3 ⊢ Ⅎ𝑥𝐶
2	1	ax-gen 1797	. 2 ⊢ ∀𝑦Ⅎ𝑥𝐶
3		csbnestgfw 4419	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝐶) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶)
4	2, 3	mpan2 689	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶)

Syntax hints:   → wi 4  ∀wal 1539   = wceq 1541   ∈ wcel 2106  Ⅎwnfc 2883  ⦋csb 3893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-v 3476  df-sbc 3778  df-csb 3894

This theorem is referenced by:  disjxpin  31814  poimirlem24  36507  cdleme31snd  39252  cdlemeg46c  39379