Theorem sbco2

Description: A composition law for substitution. For versions requiring fewer axioms, but more disjoint variable conditions, see sbco2v and sbco2vv . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 30-Jun-1994) (Revised by Mario Carneiro, 6-Oct-2016) (Proof shortened by Wolf Lammen, 17-Sep-2018) (New usage is discouraged.)

	Ref	Expression
Hypothesis	sbco2.1	⊢ Ⅎ 𝑧 𝜑
Assertion	sbco2	⊢ ( [ 𝑦 / 𝑧 ] [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		sbco2.1	⊢ Ⅎ 𝑧 𝜑
2		sbequ12	⊢ ( 𝑧 = 𝑦 → ( [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑧 ] [ 𝑧 / 𝑥 ] 𝜑 ) )
3		sbequ	⊢ ( 𝑧 = 𝑦 → ( [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) )
4	2 3	bitr3d	⊢ ( 𝑧 = 𝑦 → ( [ 𝑦 / 𝑧 ] [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) )
5	4	sps	⊢ ( ∀ 𝑧 𝑧 = 𝑦 → ( [ 𝑦 / 𝑧 ] [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) )
6		nfnae	⊢ Ⅎ 𝑧 ¬ ∀ 𝑧 𝑧 = 𝑦
7	1	nfsb4	⊢ ( ¬ ∀ 𝑧 𝑧 = 𝑦 → Ⅎ 𝑧 [ 𝑦 / 𝑥 ] 𝜑 )
8	3	a1i	⊢ ( ¬ ∀ 𝑧 𝑧 = 𝑦 → ( 𝑧 = 𝑦 → ( [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) ) )
9	6 7 8	sbied	⊢ ( ¬ ∀ 𝑧 𝑧 = 𝑦 → ( [ 𝑦 / 𝑧 ] [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) )
10	5 9	pm2.61i	⊢ ( [ 𝑦 / 𝑧 ] [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 )