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

Metamath Proof Explorer

Theorem sbco4OLD

Description: Obsolete version of sbco4 as of 3-Sep-2025. (Contributed by Jim Kingdon, 25-Sep-2018) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion sbco4OLD ( [ 𝑦 / 𝑢 ] [ 𝑥 / 𝑣 ] [ 𝑢 / 𝑥 ] [ 𝑣 / 𝑦 ] 𝜑 ↔ [ 𝑥 / 𝑤 ] [ 𝑦 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 )

Proof

Step Hyp Ref Expression
1 sbcom2 ⊢ ( [ 𝑥 / 𝑣 ] [ 𝑦 / 𝑢 ] [ 𝑢 / 𝑥 ] [ 𝑣 / 𝑦 ] 𝜑 ↔ [ 𝑦 / 𝑢 ] [ 𝑥 / 𝑣 ] [ 𝑢 / 𝑥 ] [ 𝑣 / 𝑦 ] 𝜑 )
2 sbco2vv ⊢ ( [ 𝑦 / 𝑢 ] [ 𝑢 / 𝑥 ] [ 𝑣 / 𝑦 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] [ 𝑣 / 𝑦 ] 𝜑 )
3 2 sbbii ⊢ ( [ 𝑥 / 𝑣 ] [ 𝑦 / 𝑢 ] [ 𝑢 / 𝑥 ] [ 𝑣 / 𝑦 ] 𝜑 ↔ [ 𝑥 / 𝑣 ] [ 𝑦 / 𝑥 ] [ 𝑣 / 𝑦 ] 𝜑 )
4 1 3 bitr3i ⊢ ( [ 𝑦 / 𝑢 ] [ 𝑥 / 𝑣 ] [ 𝑢 / 𝑥 ] [ 𝑣 / 𝑦 ] 𝜑 ↔ [ 𝑥 / 𝑣 ] [ 𝑦 / 𝑥 ] [ 𝑣 / 𝑦 ] 𝜑 )
5 sbco4lem ⊢ ( [ 𝑥 / 𝑣 ] [ 𝑦 / 𝑥 ] [ 𝑣 / 𝑦 ] 𝜑 ↔ [ 𝑥 / 𝑡 ] [ 𝑦 / 𝑥 ] [ 𝑡 / 𝑦 ] 𝜑 )
6 sbco4lem ⊢ ( [ 𝑥 / 𝑡 ] [ 𝑦 / 𝑥 ] [ 𝑡 / 𝑦 ] 𝜑 ↔ [ 𝑥 / 𝑤 ] [ 𝑦 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 )
7 4 5 6 3bitri ⊢ ( [ 𝑦 / 𝑢 ] [ 𝑥 / 𝑣 ] [ 𝑢 / 𝑥 ] [ 𝑣 / 𝑦 ] 𝜑 ↔ [ 𝑥 / 𝑤 ] [ 𝑦 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 )