Theorem rmoxfrd

Description: Transfer "at most one" restricted quantification from a variable x to another variable y contained in expression A . (Contributed by Thierry Arnoux, 7-Apr-2017) (Revised by Thierry Arnoux, 8-Oct-2017)

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

Proof

Step	Hyp	Ref	Expression
1		rmoxfrd.1	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) → 𝐴 ∈ 𝐵 )
2		rmoxfrd.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ∃! 𝑦 ∈ 𝐶 𝑥 = 𝐴 )
3		rmoxfrd.3	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
4		reurex	⊢ ( ∃! 𝑦 ∈ 𝐶 𝑥 = 𝐴 → ∃ 𝑦 ∈ 𝐶 𝑥 = 𝐴 )
5	2 4	syl	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ∃ 𝑦 ∈ 𝐶 𝑥 = 𝐴 )
6	1 5 3	rexxfrd	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐵 𝜓 ↔ ∃ 𝑦 ∈ 𝐶 𝜒 ) )
7		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐵 𝜓 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) )
8		df-rex	⊢ ( ∃ 𝑦 ∈ 𝐶 𝜒 ↔ ∃ 𝑦 ( 𝑦 ∈ 𝐶 ∧ 𝜒 ) )
9	6 7 8	3bitr3g	⊢ ( 𝜑 → ( ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) ↔ ∃ 𝑦 ( 𝑦 ∈ 𝐶 ∧ 𝜒 ) ) )
10	1 2 3	reuxfr1d	⊢ ( 𝜑 → ( ∃! 𝑥 ∈ 𝐵 𝜓 ↔ ∃! 𝑦 ∈ 𝐶 𝜒 ) )
11		df-reu	⊢ ( ∃! 𝑥 ∈ 𝐵 𝜓 ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) )
12		df-reu	⊢ ( ∃! 𝑦 ∈ 𝐶 𝜒 ↔ ∃! 𝑦 ( 𝑦 ∈ 𝐶 ∧ 𝜒 ) )
13	10 11 12	3bitr3g	⊢ ( 𝜑 → ( ∃! 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) ↔ ∃! 𝑦 ( 𝑦 ∈ 𝐶 ∧ 𝜒 ) ) )
14	9 13	imbi12d	⊢ ( 𝜑 → ( ( ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) → ∃! 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) ) ↔ ( ∃ 𝑦 ( 𝑦 ∈ 𝐶 ∧ 𝜒 ) → ∃! 𝑦 ( 𝑦 ∈ 𝐶 ∧ 𝜒 ) ) ) )
15		moeu	⊢ ( ∃* 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) ↔ ( ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) → ∃! 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) ) )
16		moeu	⊢ ( ∃* 𝑦 ( 𝑦 ∈ 𝐶 ∧ 𝜒 ) ↔ ( ∃ 𝑦 ( 𝑦 ∈ 𝐶 ∧ 𝜒 ) → ∃! 𝑦 ( 𝑦 ∈ 𝐶 ∧ 𝜒 ) ) )
17	14 15 16	3bitr4g	⊢ ( 𝜑 → ( ∃* 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) ↔ ∃* 𝑦 ( 𝑦 ∈ 𝐶 ∧ 𝜒 ) ) )
18		df-rmo	⊢ ( ∃* 𝑥 ∈ 𝐵 𝜓 ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) )
19		df-rmo	⊢ ( ∃* 𝑦 ∈ 𝐶 𝜒 ↔ ∃* 𝑦 ( 𝑦 ∈ 𝐶 ∧ 𝜒 ) )
20	17 18 19	3bitr4g	⊢ ( 𝜑 → ( ∃* 𝑥 ∈ 𝐵 𝜓 ↔ ∃* 𝑦 ∈ 𝐶 𝜒 ) )