Theorem sbcnestgfw 4378

Description: Nest the composition of two substitutions. Version of sbcnestgf 4383 with a disjoint variable condition, which does not require ax-13 2401. (Contributed by Mario Carneiro, 11-Nov-2016.) Avoid ax-13 2401. (Revised by GG, 26-Jan-2024.)

Assertion

Ref	Expression
sbcnestgfw	⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑))

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem sbcnestgfw

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq 3740	. . . . 5 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑))
2		csbeq1 3849	. . . . . 6 ⊢ (𝑧 = 𝐴 → ⦋𝑧 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
3	2	sbceq1d 3743	. . . . 5 ⊢ (𝑧 = 𝐴 → ([⦋𝑧 / 𝑥⦌𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑))
4	1, 3	bibi12d 348	. . . 4 ⊢ (𝑧 = 𝐴 → (([𝑧 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝑧 / 𝑥⦌𝐵 / 𝑦]𝜑) ↔ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑)))
5	4	imbi2d 343	. . 3 ⊢ (𝑧 = 𝐴 → ((∀𝑦Ⅎ𝑥𝜑 → ([𝑧 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝑧 / 𝑥⦌𝐵 / 𝑦]𝜑)) ↔ (∀𝑦Ⅎ𝑥𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑))))
6		vex 3454	. . . . 5 ⊢ 𝑧 ∈ V
7	6	a1i 11	. . . 4 ⊢ (∀𝑦Ⅎ𝑥𝜑 → 𝑧 ∈ V)
8		csbeq1a 3860	. . . . . 6 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
9	8	sbceq1d 3743	. . . . 5 ⊢ (𝑥 = 𝑧 → ([𝐵 / 𝑦]𝜑 ↔ [⦋𝑧 / 𝑥⦌𝐵 / 𝑦]𝜑))
10	9	adantl 487	. . . 4 ⊢ ((∀𝑦Ⅎ𝑥𝜑 ∧ 𝑥 = 𝑧) → ([𝐵 / 𝑦]𝜑 ↔ [⦋𝑧 / 𝑥⦌𝐵 / 𝑦]𝜑))
11		nfnf1 2190	. . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑
12	11	nfal 2353	. . . 4 ⊢ Ⅎ𝑥∀𝑦Ⅎ𝑥𝜑
13		nfa1 2187	. . . . 5 ⊢ Ⅎ𝑦∀𝑦Ⅎ𝑥𝜑
14		nfcsb1v 3870	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵
15	14	a1i 11	. . . . 5 ⊢ (∀𝑦Ⅎ𝑥𝜑 → Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵)
16		sp 2219	. . . . 5 ⊢ (∀𝑦Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑)
17	13, 15, 16	nfsbcdw 3759	. . . 4 ⊢ (∀𝑦Ⅎ𝑥𝜑 → Ⅎ𝑥[⦋𝑧 / 𝑥⦌𝐵 / 𝑦]𝜑)
18	7, 10, 12, 17	sbciedf 3780	. . 3 ⊢ (∀𝑦Ⅎ𝑥𝜑 → ([𝑧 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝑧 / 𝑥⦌𝐵 / 𝑦]𝜑))
19	5, 18	vtoclg 3517	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑦Ⅎ𝑥𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑)))
20	19	imp 412	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567   = wceq 1569  Ⅎwnf 1815   ∈ wcel 2144  Ⅎwnfc 2907  Vcvv 3450  [wsbc 3738  ⦋csb 3846

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-v 3452  df-sbc 3739  df-csb 3847

This theorem is used by:  csbnestgfw  4379  sbcnestgw  4380