Theorem cbvsbdavw2

Description: Change bound variable in proper substitution. General version of cbvsbdavw . Deduction form. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypotheses	cbvsbdavw2.1	⊢ ( 𝜑 → 𝑧 = 𝑤 )
	cbvsbdavw2.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	cbvsbdavw2	⊢ ( 𝜑 → ( [ 𝑧 / 𝑥 ] 𝜓 ↔ [ 𝑤 / 𝑦 ] 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		cbvsbdavw2.1	⊢ ( 𝜑 → 𝑧 = 𝑤 )
2		cbvsbdavw2.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
3		equequ2	⊢ ( 𝑧 = 𝑤 → ( 𝑡 = 𝑧 ↔ 𝑡 = 𝑤 ) )
4	1 3	syl	⊢ ( 𝜑 → ( 𝑡 = 𝑧 ↔ 𝑡 = 𝑤 ) )
5		equequ1	⊢ ( 𝑥 = 𝑦 → ( 𝑥 = 𝑡 ↔ 𝑦 = 𝑡 ) )
6	5	adantl	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝑥 = 𝑡 ↔ 𝑦 = 𝑡 ) )
7	6 2	imbi12d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( ( 𝑥 = 𝑡 → 𝜓 ) ↔ ( 𝑦 = 𝑡 → 𝜒 ) ) )
8	7	cbvaldvaw	⊢ ( 𝜑 → ( ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜓 ) ↔ ∀ 𝑦 ( 𝑦 = 𝑡 → 𝜒 ) ) )
9	4 8	imbi12d	⊢ ( 𝜑 → ( ( 𝑡 = 𝑧 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜓 ) ) ↔ ( 𝑡 = 𝑤 → ∀ 𝑦 ( 𝑦 = 𝑡 → 𝜒 ) ) ) )
10	9	albidv	⊢ ( 𝜑 → ( ∀ 𝑡 ( 𝑡 = 𝑧 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜓 ) ) ↔ ∀ 𝑡 ( 𝑡 = 𝑤 → ∀ 𝑦 ( 𝑦 = 𝑡 → 𝜒 ) ) ) )
11		df-sb	⊢ ( [ 𝑧 / 𝑥 ] 𝜓 ↔ ∀ 𝑡 ( 𝑡 = 𝑧 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜓 ) ) )
12		df-sb	⊢ ( [ 𝑤 / 𝑦 ] 𝜒 ↔ ∀ 𝑡 ( 𝑡 = 𝑤 → ∀ 𝑦 ( 𝑦 = 𝑡 → 𝜒 ) ) )
13	10 11 12	3bitr4g	⊢ ( 𝜑 → ( [ 𝑧 / 𝑥 ] 𝜓 ↔ [ 𝑤 / 𝑦 ] 𝜒 ) )