Theorem ralxfrd2

Description: Transfer universal quantification from a variable x to another variable y contained in expression A . Variant of ralxfrd . (Contributed by Alexander van der Vekens, 25-Apr-2018)

	Ref	Expression
Hypotheses	ralxfrd2.1	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) → 𝐴 ∈ 𝐵 )
	ralxfrd2.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ∃ 𝑦 ∈ 𝐶 𝑥 = 𝐴 )
	ralxfrd2.3	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	ralxfrd2	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐵 𝜓 ↔ ∀ 𝑦 ∈ 𝐶 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		ralxfrd2.1	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) → 𝐴 ∈ 𝐵 )
2		ralxfrd2.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ∃ 𝑦 ∈ 𝐶 𝑥 = 𝐴 )
3		ralxfrd2.3	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
4	3	3expa	⊢ ( ( ( 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	ad4ant134	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐶 ) ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
9	8	exbiri	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐶 ) → ( 𝑥 = 𝐴 → ( 𝜒 → 𝜓 ) ) )
10	9	impcomd	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 ∈ 𝐶 ) → ( ( 𝜒 ∧ 𝑥 = 𝐴 ) → 𝜓 ) )
11	10	rexlimdva	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( ∃ 𝑦 ∈ 𝐶 ( 𝜒 ∧ 𝑥 = 𝐴 ) → 𝜓 ) )
12	7 11	syl5	⊢ ( ( 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 ) )