Theorem sbcnestgw 4379

Description: Nest the composition of two substitutions. Version of sbcnestg 4384 with a disjoint variable condition, which does not require ax-13 2405. (Contributed by NM, 27-Nov-2005.) Avoid ax-13 2405. (Revised by GG, 26-Jan-2024.)

Assertion

Ref	Expression
sbcnestgw	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑))

Distinct variable groups:   𝑥,𝑦   𝜑,𝑥

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

Proof of Theorem sbcnestgw

Step	Hyp	Ref	Expression
1		nfv 1936	. . 3 ⊢ Ⅎ𝑥𝜑
2	1	ax-gen 1817	. 2 ⊢ ∀𝑦Ⅎ𝑥𝜑
3		sbcnestgfw 4377	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑))
4	2, 3	mpan2 701	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1560  Ⅎwnf 1805   ∈ wcel 2144  [wsbc 3746  ⦋csb 3854

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-ex 1802  df-nf 1806  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-v 3458  df-sbc 3747  df-csb 3855

This theorem is referenced by:  sbcco3gw  4381  sbcop  5459