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Metamath Proof Explorer

Theorem cbvabw

Description: Rule used to change bound variables, using implicit substitution. Version of cbvab with a disjoint variable condition, which does not require ax-10 , ax-13 . (Contributed by Andrew Salmon, 11-Jul-2011) (Revised by GG, 23-May-2024)

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

Proof

Step	Hyp	Ref	Expression
1		cbvabw.1	⊢ Ⅎ 𝑦 𝜑
2		cbvabw.2	⊢ Ⅎ 𝑥 𝜓
3		cbvabw.3	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
4	1 2 3	cbvsbvf	⊢ ( [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝑧 / 𝑦 ] 𝜓 )
5		df-clab	⊢ ( 𝑧 ∈ { 𝑥 ∣ 𝜑 } ↔ [ 𝑧 / 𝑥 ] 𝜑 )
6		df-clab	⊢ ( 𝑧 ∈ { 𝑦 ∣ 𝜓 } ↔ [ 𝑧 / 𝑦 ] 𝜓 )
7	4 5 6	3bitr4i	⊢ ( 𝑧 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝑧 ∈ { 𝑦 ∣ 𝜓 } )
8	7	eqriv	⊢ { 𝑥 ∣ 𝜑 } = { 𝑦 ∣ 𝜓 }