Theorem cbvab

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Usage of the weaker cbvabw and cbvabv are preferred. (Contributed by Andrew Salmon, 11-Jul-2011) (Proof shortened by Wolf Lammen, 16-Nov-2019) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbvab.1	⊢ Ⅎ 𝑦 𝜑
	cbvab.2	⊢ Ⅎ 𝑥 𝜓
	cbvab.3	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	cbvab	⊢ { 𝑥 ∣ 𝜑 } = { 𝑦 ∣ 𝜓 }

Proof

Step	Hyp	Ref	Expression
1		cbvab.1	⊢ Ⅎ 𝑦 𝜑
2		cbvab.2	⊢ Ⅎ 𝑥 𝜓
3		cbvab.3	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
4	1	sbco2	⊢ ( [ 𝑧 / 𝑦 ] [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑧 / 𝑥 ] 𝜑 )
5	2 3	sbie	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 )
6	5	sbbii	⊢ ( [ 𝑧 / 𝑦 ] [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑧 / 𝑦 ] 𝜓 )
7	4 6	bitr3i	⊢ ( [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝑧 / 𝑦 ] 𝜓 )
8		df-clab	⊢ ( 𝑧 ∈ { 𝑥 ∣ 𝜑 } ↔ [ 𝑧 / 𝑥 ] 𝜑 )
9		df-clab	⊢ ( 𝑧 ∈ { 𝑦 ∣ 𝜓 } ↔ [ 𝑧 / 𝑦 ] 𝜓 )
10	7 8 9	3bitr4i	⊢ ( 𝑧 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝑧 ∈ { 𝑦 ∣ 𝜓 } )
11	10	eqriv	⊢ { 𝑥 ∣ 𝜑 } = { 𝑦 ∣ 𝜓 }