Theorem csbnestgw 4416

Description: Nest the composition of two substitutions. Version of csbnestg 4421 with a disjoint variable condition, which does not require ax-13 2365. (Contributed by NM, 23-Nov-2005.) Avoid ax-13 2365. (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 2897	. . 3 ⊢ Ⅎ𝑥𝐶
2	1	ax-gen 1789	. 2 ⊢ ∀𝑦Ⅎ𝑥𝐶
3		csbnestgfw 4414	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝐶) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶)
4	2, 3	mpan2 688	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶)

Syntax hints:   → wi 4  ∀wal 1531   = wceq 1533   ∈ wcel 2098  Ⅎwnfc 2877  ⦋csb 3888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-v 3470  df-sbc 3773  df-csb 3889

This theorem is referenced by:  disjxpin  32328  poimirlem24  37025  cdleme31snd  39770  cdlemeg46c  39897