Theorem ralxfrd

Description: Transfer universal quantification from a variable x to another variable y contained in expression A . (Contributed by NM, 15-Aug-2014) (Proof shortened by Mario Carneiro, 19-Nov-2016) (Proof shortened by JJ, 7-Aug-2021)

	Ref	Expression
Hypotheses	ralxfrd.1	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) → 𝐴 ∈ 𝐵 )
	ralxfrd.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ∃ 𝑦 ∈ 𝐶 𝑥 = 𝐴 )
	ralxfrd.3	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	ralxfrd	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐵 𝜓 ↔ ∀ 𝑦 ∈ 𝐶 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		ralxfrd.1	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) → 𝐴 ∈ 𝐵 )
2		ralxfrd.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ∃ 𝑦 ∈ 𝐶 𝑥 = 𝐴 )
3		ralxfrd.3	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
4	3	adantlr	⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
5	1 4	rspcdv	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) → ( ∀ 𝑥 ∈ 𝐵 𝜓 → 𝜒 ) )
6	5	ralrimdva	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐵 𝜓 → ∀ 𝑦 ∈ 𝐶 𝜒 ) )
7		r19.29	⊢ ( ( ∀ 𝑦 ∈ 𝐶 𝜒 ∧ ∃ 𝑦 ∈ 𝐶 𝑥 = 𝐴 ) → ∃ 𝑦 ∈ 𝐶 ( 𝜒 ∧ 𝑥 = 𝐴 ) )
8	3	exbiri	⊢ ( 𝜑 → ( 𝑥 = 𝐴 → ( 𝜒 → 𝜓 ) ) )
9	8	impcomd	⊢ ( 𝜑 → ( ( 𝜒 ∧ 𝑥 = 𝐴 ) → 𝜓 ) )
10	9	rexlimdvw	⊢ ( 𝜑 → ( ∃ 𝑦 ∈ 𝐶 ( 𝜒 ∧ 𝑥 = 𝐴 ) → 𝜓 ) )
11	7 10	syl5	⊢ ( 𝜑 → ( ( ∀ 𝑦 ∈ 𝐶 𝜒 ∧ ∃ 𝑦 ∈ 𝐶 𝑥 = 𝐴 ) → 𝜓 ) )
12	11	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( ( ∀ 𝑦 ∈ 𝐶 𝜒 ∧ ∃ 𝑦 ∈ 𝐶 𝑥 = 𝐴 ) → 𝜓 ) )
13	2 12	mpan2d	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( ∀ 𝑦 ∈ 𝐶 𝜒 → 𝜓 ) )
14	13	ralrimdva	⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝐶 𝜒 → ∀ 𝑥 ∈ 𝐵 𝜓 ) )
15	6 14	impbid	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐵 𝜓 ↔ ∀ 𝑦 ∈ 𝐶 𝜒 ) )