Theorem cbvals

Description: Rule used to change bound variables, using implicit substitution. (Contributed by David A. Wheeler, 12-Jul-2026)

	Ref	Expression
Hypotheses	cbvals.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜒 ) )
	cbvals.2	⊢ ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜃 ) )
Assertion	cbvals	⊢ ( ∀∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ∀∃ 𝑦 ( 𝜒 → 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		cbvals.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜒 ) )
2		cbvals.2	⊢ ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜃 ) )
3	1 2	imbi12d	⊢ ( 𝑥 = 𝑦 → ( ( 𝜑 → 𝜓 ) ↔ ( 𝜒 → 𝜃 ) ) )
4	3	cbvalvw	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ∀ 𝑦 ( 𝜒 → 𝜃 ) )
5	1	cbvexvw	⊢ ( ∃ 𝑥 𝜑 ↔ ∃ 𝑦 𝜒 )
6	4 5	anbi12i	⊢ ( ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ∧ ∃ 𝑥 𝜑 ) ↔ ( ∀ 𝑦 ( 𝜒 → 𝜃 ) ∧ ∃ 𝑦 𝜒 ) )
7		df-als	⊢ ( ∀∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ∧ ∃ 𝑥 𝜑 ) )
8		df-als	⊢ ( ∀∃ 𝑦 ( 𝜒 → 𝜃 ) ↔ ( ∀ 𝑦 ( 𝜒 → 𝜃 ) ∧ ∃ 𝑦 𝜒 ) )
9	6 7 8	3bitr4i	⊢ ( ∀∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ∀∃ 𝑦 ( 𝜒 → 𝜃 ) )