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Metamath Proof Explorer

Theorem sbco3

Description: A composition law for substitution. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 2-Jun-1993) (Proof shortened by Wolf Lammen, 18-Sep-2018) (New usage is discouraged.)

Ref Expression
Assertion sbco3 ( [ 𝑧 / 𝑦 ] [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑧 / 𝑥 ] [ 𝑥 / 𝑦 ] 𝜑 )

Proof

Step Hyp Ref Expression
1 drsb1 ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑧 / 𝑥 ] [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑧 / 𝑦 ] [ 𝑦 / 𝑥 ] 𝜑 ) )
2 nfae ⊢ Ⅎ 𝑥 ∀ 𝑥 𝑥 = 𝑦
3 sbequ12a ⊢ ( 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑥 / 𝑦 ] 𝜑 ) )
4 3 sps ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑥 / 𝑦 ] 𝜑 ) )
5 2 4 sbbid ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑧 / 𝑥 ] [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑧 / 𝑥 ] [ 𝑥 / 𝑦 ] 𝜑 ) )
6 1 5 bitr3d ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑧 / 𝑦 ] [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑧 / 𝑥 ] [ 𝑥 / 𝑦 ] 𝜑 ) )
7 nfnae ⊢ Ⅎ 𝑦 ¬ ∀ 𝑥 𝑥 = 𝑦
8 nfnae ⊢ Ⅎ 𝑥 ¬ ∀ 𝑥 𝑥 = 𝑦
9 nfsb2 ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 )
10 7 8 9 sbco2d ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑧 / 𝑥 ] [ 𝑥 / 𝑦 ] [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑧 / 𝑦 ] [ 𝑦 / 𝑥 ] 𝜑 ) )
11 sbco ⊢ ( [ 𝑥 / 𝑦 ] [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑥 / 𝑦 ] 𝜑 )
12 11 sbbii ⊢ ( [ 𝑧 / 𝑥 ] [ 𝑥 / 𝑦 ] [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑧 / 𝑥 ] [ 𝑥 / 𝑦 ] 𝜑 )
13 10 12 bitr3di ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( [ 𝑧 / 𝑦 ] [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑧 / 𝑥 ] [ 𝑥 / 𝑦 ] 𝜑 ) )
14 6 13 pm2.61i ⊢ ( [ 𝑧 / 𝑦 ] [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑧 / 𝑥 ] [ 𝑥 / 𝑦 ] 𝜑 )