Theorem cbvraldva2

Description: Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017)

	Ref	Expression
Hypotheses	cbvraldva2.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
	cbvraldva2.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐴 = 𝐵 )
Assertion	cbvraldva2	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑦 ∈ 𝐵 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		cbvraldva2.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
2		cbvraldva2.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐴 = 𝐵 )
3		simpr	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝑥 = 𝑦 )
4	3 2	eleq12d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵 ) )
5	4 1	imbi12d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( ( 𝑥 ∈ 𝐴 → 𝜓 ) ↔ ( 𝑦 ∈ 𝐵 → 𝜒 ) ) )
6	5	expcom	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → ( ( 𝑥 ∈ 𝐴 → 𝜓 ) ↔ ( 𝑦 ∈ 𝐵 → 𝜒 ) ) ) )
7	6	pm5.74d	⊢ ( 𝑥 = 𝑦 → ( ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝜓 ) ) ↔ ( 𝜑 → ( 𝑦 ∈ 𝐵 → 𝜒 ) ) ) )
8	7	cbvalvw	⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝜓 ) ) ↔ ∀ 𝑦 ( 𝜑 → ( 𝑦 ∈ 𝐵 → 𝜒 ) ) )
9		19.21v	⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝜓 ) ) ↔ ( 𝜑 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) ) )
10		19.21v	⊢ ( ∀ 𝑦 ( 𝜑 → ( 𝑦 ∈ 𝐵 → 𝜒 ) ) ↔ ( 𝜑 → ∀ 𝑦 ( 𝑦 ∈ 𝐵 → 𝜒 ) ) )
11	8 9 10	3bitr3i	⊢ ( ( 𝜑 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) ) ↔ ( 𝜑 → ∀ 𝑦 ( 𝑦 ∈ 𝐵 → 𝜒 ) ) )
12	11	pm5.74ri	⊢ ( 𝜑 → ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) ↔ ∀ 𝑦 ( 𝑦 ∈ 𝐵 → 𝜒 ) ) )
13		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) )
14		df-ral	⊢ ( ∀ 𝑦 ∈ 𝐵 𝜒 ↔ ∀ 𝑦 ( 𝑦 ∈ 𝐵 → 𝜒 ) )
15	12 13 14	3bitr4g	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑦 ∈ 𝐵 𝜒 ) )