Theorem sbccom

Description: Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005) (Proof shortened by Mario Carneiro, 18-Oct-2016)

	Ref	Expression
Assertion	sbccom	⊢ ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ↔ [ 𝐵 / 𝑦 ] [ 𝐴 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		sbccomlem	⊢ ( [ 𝐴 / 𝑧 ] [ 𝐵 / 𝑤 ] [ 𝑤 / 𝑦 ] [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝐵 / 𝑤 ] [ 𝐴 / 𝑧 ] [ 𝑤 / 𝑦 ] [ 𝑧 / 𝑥 ] 𝜑 )
2		sbccomlem	⊢ ( [ 𝑤 / 𝑦 ] [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 )
3	2	sbcbii	⊢ ( [ 𝐵 / 𝑤 ] [ 𝑤 / 𝑦 ] [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝐵 / 𝑤 ] [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 )
4		sbccomlem	⊢ ( [ 𝐵 / 𝑤 ] [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ↔ [ 𝑧 / 𝑥 ] [ 𝐵 / 𝑤 ] [ 𝑤 / 𝑦 ] 𝜑 )
5	3 4	bitri	⊢ ( [ 𝐵 / 𝑤 ] [ 𝑤 / 𝑦 ] [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝑧 / 𝑥 ] [ 𝐵 / 𝑤 ] [ 𝑤 / 𝑦 ] 𝜑 )
6	5	sbcbii	⊢ ( [ 𝐴 / 𝑧 ] [ 𝐵 / 𝑤 ] [ 𝑤 / 𝑦 ] [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝐴 / 𝑧 ] [ 𝑧 / 𝑥 ] [ 𝐵 / 𝑤 ] [ 𝑤 / 𝑦 ] 𝜑 )
7		sbccomlem	⊢ ( [ 𝐴 / 𝑧 ] [ 𝑤 / 𝑦 ] [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝑤 / 𝑦 ] [ 𝐴 / 𝑧 ] [ 𝑧 / 𝑥 ] 𝜑 )
8	7	sbcbii	⊢ ( [ 𝐵 / 𝑤 ] [ 𝐴 / 𝑧 ] [ 𝑤 / 𝑦 ] [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝐵 / 𝑤 ] [ 𝑤 / 𝑦 ] [ 𝐴 / 𝑧 ] [ 𝑧 / 𝑥 ] 𝜑 )
9	1 6 8	3bitr3i	⊢ ( [ 𝐴 / 𝑧 ] [ 𝑧 / 𝑥 ] [ 𝐵 / 𝑤 ] [ 𝑤 / 𝑦 ] 𝜑 ↔ [ 𝐵 / 𝑤 ] [ 𝑤 / 𝑦 ] [ 𝐴 / 𝑧 ] [ 𝑧 / 𝑥 ] 𝜑 )
10		sbccow	⊢ ( [ 𝐴 / 𝑧 ] [ 𝑧 / 𝑥 ] [ 𝐵 / 𝑤 ] [ 𝑤 / 𝑦 ] 𝜑 ↔ [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑤 ] [ 𝑤 / 𝑦 ] 𝜑 )
11		sbccow	⊢ ( [ 𝐵 / 𝑤 ] [ 𝑤 / 𝑦 ] [ 𝐴 / 𝑧 ] [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝐵 / 𝑦 ] [ 𝐴 / 𝑧 ] [ 𝑧 / 𝑥 ] 𝜑 )
12	9 10 11	3bitr3i	⊢ ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑤 ] [ 𝑤 / 𝑦 ] 𝜑 ↔ [ 𝐵 / 𝑦 ] [ 𝐴 / 𝑧 ] [ 𝑧 / 𝑥 ] 𝜑 )
13		sbccow	⊢ ( [ 𝐵 / 𝑤 ] [ 𝑤 / 𝑦 ] 𝜑 ↔ [ 𝐵 / 𝑦 ] 𝜑 )
14	13	sbcbii	⊢ ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑤 ] [ 𝑤 / 𝑦 ] 𝜑 ↔ [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 )
15		sbccow	⊢ ( [ 𝐴 / 𝑧 ] [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜑 )
16	15	sbcbii	⊢ ( [ 𝐵 / 𝑦 ] [ 𝐴 / 𝑧 ] [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝐵 / 𝑦 ] [ 𝐴 / 𝑥 ] 𝜑 )
17	12 14 16	3bitr3i	⊢ ( [ 𝐴 / 𝑥 ] [ 𝐵 / 𝑦 ] 𝜑 ↔ [ 𝐵 / 𝑦 ] [ 𝐴 / 𝑥 ] 𝜑 )