Theorem cbvsbcdavw

Description: Change bound variable of a class substitution. Deduction form. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypothesis	cbvsbcdavw.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	cbvsbcdavw	⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ [ 𝐴 / 𝑦 ] 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		cbvsbcdavw.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
2	1	cbvabdavw	⊢ ( 𝜑 → { 𝑥 ∣ 𝜓 } = { 𝑦 ∣ 𝜒 } )
3	2	eleq2d	⊢ ( 𝜑 → ( 𝐴 ∈ { 𝑥 ∣ 𝜓 } ↔ 𝐴 ∈ { 𝑦 ∣ 𝜒 } ) )
4		df-sbc	⊢ ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝐴 ∈ { 𝑥 ∣ 𝜓 } )
5		df-sbc	⊢ ( [ 𝐴 / 𝑦 ] 𝜒 ↔ 𝐴 ∈ { 𝑦 ∣ 𝜒 } )
6	3 4 5	3bitr4g	⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ [ 𝐴 / 𝑦 ] 𝜒 ) )