Theorem sbcnestgf 4391

Description: Nest the composition of two substitutions. Usage of this theorem is discouraged because it depends on ax-13 2404. Use the weaker sbcnestgfw 4386 when possible. (Contributed by Mario Carneiro, 11-Nov-2016.) (New usage is discouraged.)

Assertion

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

Proof of Theorem sbcnestgf

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq 3746	. . . . 5 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑))
2		csbeq1 3856	. . . . . 6 ⊢ (𝑧 = 𝐴 → ⦋𝑧 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
3	2	sbceq1d 3749	. . . . 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 3459	. . . . 5 ⊢ 𝑧 ∈ V
7	6	a1i 11	. . . 4 ⊢ (∀𝑦Ⅎ𝑥𝜑 → 𝑧 ∈ V)
8		csbeq1a 3867	. . . . . 6 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
9	8	sbceq1d 3749	. . . . 5 ⊢ (𝑥 = 𝑧 → ([𝐵 / 𝑦]𝜑 ↔ [⦋𝑧 / 𝑥⦌𝐵 / 𝑦]𝜑))
10	9	adantl 486	. . . 4 ⊢ ((∀𝑦Ⅎ𝑥𝜑 ∧ 𝑥 = 𝑧) → ([𝐵 / 𝑦]𝜑 ↔ [⦋𝑧 / 𝑥⦌𝐵 / 𝑦]𝜑))
11		nfnf1 2189	. . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑
12	11	nfal 2356	. . . 4 ⊢ Ⅎ𝑥∀𝑦Ⅎ𝑥𝜑
13		nfa1 2186	. . . . 5 ⊢ Ⅎ𝑦∀𝑦Ⅎ𝑥𝜑
14		nfcsb1v 3877	. . . . . 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	nfsbcd 3768	. . . 4 ⊢ (∀𝑦Ⅎ𝑥𝜑 → Ⅎ𝑥[⦋𝑧 / 𝑥⦌𝐵 / 𝑦]𝜑)
18	7, 10, 12, 17	sbciedf 3786	. . 3 ⊢ (∀𝑦Ⅎ𝑥𝜑 → ([𝑧 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝑧 / 𝑥⦌𝐵 / 𝑦]𝜑))
19	5, 18	vtoclg 3522	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑦Ⅎ𝑥𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑)))
20	19	imp 411	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1568   = wceq 1570  Ⅎwnf 1813   ∈ wcel 2143  Ⅎwnfc 2910  Vcvv 3455  [wsbc 3744  ⦋csb 3853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-13 2404  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-v 3457  df-sbc 3745  df-csb 3854

This theorem is referenced by:  csbnestgf  4392  sbcnestg  4393