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Metamath Proof Explorer

Theorem sbnf2

Description: Two ways of expressing " x is (effectively) not free in ph ". (Contributed by Gérard Lang, 14-Nov-2013) (Revised by Mario Carneiro, 6-Oct-2016) (Proof shortened by Wolf Lammen, 22-Sep-2018) Avoid ax-13 . (Revised by Wolf Lammen, 30-Jan-2023)

Ref Expression
Assertion sbnf2 ( Ⅎ 𝑥 𝜑 ↔ ∀ 𝑦 ∀ 𝑧 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑧 / 𝑥 ] 𝜑 ) )

Proof

Step Hyp Ref Expression
1 nfv ⊢ Ⅎ 𝑦 𝜑
2 1 sb8ef ⊢ ( ∃ 𝑥 𝜑 ↔ ∃ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )
3 sb8v ⊢ ( ∀ 𝑥 𝜑 ↔ ∀ 𝑧 [ 𝑧 / 𝑥 ] 𝜑 )
4 2 3 imbi12i ⊢ ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) ↔ ( ∃ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 → ∀ 𝑧 [ 𝑧 / 𝑥 ] 𝜑 ) )
5 df-nf ⊢ ( Ⅎ 𝑥 𝜑 ↔ ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) )
6 pm11.53v ⊢ ( ∀ 𝑦 ∀ 𝑧 ( [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑧 / 𝑥 ] 𝜑 ) ↔ ( ∃ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 → ∀ 𝑧 [ 𝑧 / 𝑥 ] 𝜑 ) )
7 4 5 6 3bitr4i ⊢ ( Ⅎ 𝑥 𝜑 ↔ ∀ 𝑦 ∀ 𝑧 ( [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑧 / 𝑥 ] 𝜑 ) )
8 nfv ⊢ Ⅎ 𝑧 𝜑
9 8 sb8ef ⊢ ( ∃ 𝑥 𝜑 ↔ ∃ 𝑧 [ 𝑧 / 𝑥 ] 𝜑 )
10 sb8v ⊢ ( ∀ 𝑥 𝜑 ↔ ∀ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )
11 9 10 imbi12i ⊢ ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) ↔ ( ∃ 𝑧 [ 𝑧 / 𝑥 ] 𝜑 → ∀ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 ) )
12 pm11.53v ⊢ ( ∀ 𝑧 ∀ 𝑦 ( [ 𝑧 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] 𝜑 ) ↔ ( ∃ 𝑧 [ 𝑧 / 𝑥 ] 𝜑 → ∀ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 ) )
13 11 5 12 3bitr4i ⊢ ( Ⅎ 𝑥 𝜑 ↔ ∀ 𝑧 ∀ 𝑦 ( [ 𝑧 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] 𝜑 ) )
14 alcom ⊢ ( ∀ 𝑧 ∀ 𝑦 ( [ 𝑧 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] 𝜑 ) ↔ ∀ 𝑦 ∀ 𝑧 ( [ 𝑧 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] 𝜑 ) )
15 13 14 bitri ⊢ ( Ⅎ 𝑥 𝜑 ↔ ∀ 𝑦 ∀ 𝑧 ( [ 𝑧 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] 𝜑 ) )
16 7 15 anbi12i ⊢ ( ( Ⅎ 𝑥 𝜑 ∧ Ⅎ 𝑥 𝜑 ) ↔ ( ∀ 𝑦 ∀ 𝑧 ( [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑧 / 𝑥 ] 𝜑 ) ∧ ∀ 𝑦 ∀ 𝑧 ( [ 𝑧 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] 𝜑 ) ) )
17 pm4.24 ⊢ ( Ⅎ 𝑥 𝜑 ↔ ( Ⅎ 𝑥 𝜑 ∧ Ⅎ 𝑥 𝜑 ) )
18 2albiim ⊢ ( ∀ 𝑦 ∀ 𝑧 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑧 / 𝑥 ] 𝜑 ) ↔ ( ∀ 𝑦 ∀ 𝑧 ( [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑧 / 𝑥 ] 𝜑 ) ∧ ∀ 𝑦 ∀ 𝑧 ( [ 𝑧 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] 𝜑 ) ) )
19 16 17 18 3bitr4i ⊢ ( Ⅎ 𝑥 𝜑 ↔ ∀ 𝑦 ∀ 𝑧 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑧 / 𝑥 ] 𝜑 ) )