Theorem csbnestgw 4382

Description: Nest the composition of two substitutions. Version of csbnestg 4387 with a disjoint variable condition, which does not require ax-13 2401. (Contributed by NM, 23-Nov-2005.) Avoid ax-13 2401. (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 2922	. . 3 ⊢ Ⅎ𝑥𝐶
2	1	ax-gen 1828	. 2 ⊢ ∀𝑦Ⅎ𝑥𝐶
3		csbnestgfw 4380	. 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 2145  Ⅎwnfc 2907  ⦋csb 3847

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 2732

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 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-v 3452  df-sbc 3740  df-csb 3848

This theorem is used by:  disjxpin  33062  poimirlem24  38394  cdleme31snd  41260  cdlemeg46c  41387