This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem sbcnestgfw

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

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

Proof

Step Hyp Ref Expression
1 dfsbcq ⊢ ( 𝑧 = 𝐴 → ( [ 𝑧 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ↔ [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ) )
2 csbeq1 ⊢ ( 𝑧 = 𝐴 → ⦋ 𝑧 / 𝑥 ⦌ 𝐵 = ⦋ 𝐴 / 𝑥 ⦌ 𝐵 )
3 2 sbceq1d ⊢ ( 𝑧 = 𝐴 → ( [ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 / 𝑦 ] 𝜑 ↔ [ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 / 𝑦 ] 𝜑 ) )
4 1 3 bibi12d ⊢ ( 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 ⊢ ( 𝑧 = 𝐴 → ( ( ∀ 𝑦 Ⅎ 𝑥 𝜑 → ( [ 𝑧 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ↔ [ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 / 𝑦 ] 𝜑 ) ) ↔ ( ∀ 𝑦 Ⅎ 𝑥 𝜑 → ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ↔ [ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 / 𝑦 ] 𝜑 ) ) ) )
6 vex ⊢ 𝑧 ∈ V
7 6 a1i ⊢ ( ∀ 𝑦 Ⅎ 𝑥 𝜑 → 𝑧 ∈ V )
8 csbeq1a ⊢ ( 𝑥 = 𝑧 → 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 )
9 8 sbceq1d ⊢ ( 𝑥 = 𝑧 → ( [ 𝐵 / 𝑦 ] 𝜑 ↔ [ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 / 𝑦 ] 𝜑 ) )
10 9 adantl ⊢ ( ( ∀ 𝑦 Ⅎ 𝑥 𝜑 ∧ 𝑥 = 𝑧 ) → ( [ 𝐵 / 𝑦 ] 𝜑 ↔ [ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 / 𝑦 ] 𝜑 ) )
11 nfnf1 ⊢ Ⅎ 𝑥 Ⅎ 𝑥 𝜑
12 11 nfal ⊢ Ⅎ 𝑥 ∀ 𝑦 Ⅎ 𝑥 𝜑
13 nfa1 ⊢ Ⅎ 𝑦 ∀ 𝑦 Ⅎ 𝑥 𝜑
14 nfcsb1v ⊢ Ⅎ 𝑥 ⦋ 𝑧 / 𝑥 ⦌ 𝐵
15 14 a1i ⊢ ( ∀ 𝑦 Ⅎ 𝑥 𝜑 → Ⅎ 𝑥 ⦋ 𝑧 / 𝑥 ⦌ 𝐵 )
16 sp ⊢ ( ∀ 𝑦 Ⅎ 𝑥 𝜑 → Ⅎ 𝑥 𝜑 )
17 13 15 16 nfsbcdw ⊢ ( ∀ 𝑦 Ⅎ 𝑥 𝜑 → Ⅎ 𝑥 [ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 / 𝑦 ] 𝜑 )
18 7 10 12 17 sbciedf ⊢ ( ∀ 𝑦 Ⅎ 𝑥 𝜑 → ( [ 𝑧 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ↔ [ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 / 𝑦 ] 𝜑 ) )
19 5 18 vtoclg ⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑦 Ⅎ 𝑥 𝜑 → ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ↔ [ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 / 𝑦 ] 𝜑 ) ) )
20 19 imp ⊢ ( ( 𝐴 ∈ 𝑉 ∧ ∀ 𝑦 Ⅎ 𝑥 𝜑 ) → ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ↔ [ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 / 𝑦 ] 𝜑 ) )