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 )