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Metamath Proof Explorer

Theorem sbnfOLD

Description: Obsolete version of sbnf as of 2-May-2025. (Contributed by BJ, 2-May-2019) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 sbim ⊢ ( [ 𝑧 / 𝑦 ] ( 𝜑 → ∀ 𝑥 𝜑 ) ↔ ( [ 𝑧 / 𝑦 ] 𝜑 → [ 𝑧 / 𝑦 ] ∀ 𝑥 𝜑 ) )
2 sbal ⊢ ( [ 𝑧 / 𝑦 ] ∀ 𝑥 𝜑 ↔ ∀ 𝑥 [ 𝑧 / 𝑦 ] 𝜑 )
3 2 imbi2i ⊢ ( ( [ 𝑧 / 𝑦 ] 𝜑 → [ 𝑧 / 𝑦 ] ∀ 𝑥 𝜑 ) ↔ ( [ 𝑧 / 𝑦 ] 𝜑 → ∀ 𝑥 [ 𝑧 / 𝑦 ] 𝜑 ) )
4 1 3 bitri ⊢ ( [ 𝑧 / 𝑦 ] ( 𝜑 → ∀ 𝑥 𝜑 ) ↔ ( [ 𝑧 / 𝑦 ] 𝜑 → ∀ 𝑥 [ 𝑧 / 𝑦 ] 𝜑 ) )
5 4 albii ⊢ ( ∀ 𝑥 [ 𝑧 / 𝑦 ] ( 𝜑 → ∀ 𝑥 𝜑 ) ↔ ∀ 𝑥 ( [ 𝑧 / 𝑦 ] 𝜑 → ∀ 𝑥 [ 𝑧 / 𝑦 ] 𝜑 ) )
6 nf5 ⊢ ( Ⅎ 𝑥 𝜑 ↔ ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) )
7 6 sbbii ⊢ ( [ 𝑧 / 𝑦 ] Ⅎ 𝑥 𝜑 ↔ [ 𝑧 / 𝑦 ] ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) )
8 sbal ⊢ ( [ 𝑧 / 𝑦 ] ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) ↔ ∀ 𝑥 [ 𝑧 / 𝑦 ] ( 𝜑 → ∀ 𝑥 𝜑 ) )
9 7 8 bitri ⊢ ( [ 𝑧 / 𝑦 ] Ⅎ 𝑥 𝜑 ↔ ∀ 𝑥 [ 𝑧 / 𝑦 ] ( 𝜑 → ∀ 𝑥 𝜑 ) )
10 nf5 ⊢ ( Ⅎ 𝑥 [ 𝑧 / 𝑦 ] 𝜑 ↔ ∀ 𝑥 ( [ 𝑧 / 𝑦 ] 𝜑 → ∀ 𝑥 [ 𝑧 / 𝑦 ] 𝜑 ) )
11 5 9 10 3bitr4i ⊢ ( [ 𝑧 / 𝑦 ] Ⅎ 𝑥 𝜑 ↔ Ⅎ 𝑥 [ 𝑧 / 𝑦 ] 𝜑 )