Theorem sbcnestgw 4354

Description: Nest the composition of two substitutions. Version of sbcnestg 4359 with a disjoint variable condition, which does not require ax-13 2382. (Contributed by NM, 27-Nov-2005.) Avoid ax-13 2382. (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 1922	. . 3 ⊢ Ⅎ𝑥𝜑
2	1	ax-gen 1803	. 2 ⊢ ∀𝑦Ⅎ𝑥𝜑
3		sbcnestgfw 4352	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑))
4	2, 3	mpan2 698	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1546  Ⅎwnf 1791   ∈ wcel 2121  [wsbc 3725  ⦋csb 3833

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-ex 1788  df-nf 1792  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-v 3435  df-sbc 3726  df-csb 3834

This theorem is referenced by:  sbcco3gw  4356  sbcop  5432