Theorem sbcnestgw 4415

Description: Nest the composition of two substitutions. Version of sbcnestg 4420 with a disjoint variable condition, which does not require ax-13 2365. (Contributed by NM, 27-Nov-2005.) Avoid ax-13 2365. (Revised by Gino Giotto, 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 1909	. . 3 ⊢ Ⅎ𝑥𝜑
2	1	ax-gen 1789	. 2 ⊢ ∀𝑦Ⅎ𝑥𝜑
3		sbcnestgfw 4413	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑))
4	2, 3	mpan2 688	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1531  Ⅎwnf 1777   ∈ wcel 2098  [wsbc 3772  ⦋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:  sbcco3gw  4417  sbcop  5482