Theorem reuxfr1d

Description: Transfer existential uniqueness from a variable x to another variable y contained in expression A . Cf. reuxfr1ds . (Contributed by Thierry Arnoux, 7-Apr-2017)

	Ref	Expression
Hypotheses	reuxfr1d.1	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) → 𝐴 ∈ 𝐵 )
	reuxfr1d.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ∃! 𝑦 ∈ 𝐶 𝑥 = 𝐴 )
	reuxfr1d.3	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	reuxfr1d	⊢ ( 𝜑 → ( ∃! 𝑥 ∈ 𝐵 𝜓 ↔ ∃! 𝑦 ∈ 𝐶 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		reuxfr1d.1	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐶 ) → 𝐴 ∈ 𝐵 )
2		reuxfr1d.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ∃! 𝑦 ∈ 𝐶 𝑥 = 𝐴 )
3		reuxfr1d.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	5	biantrurd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝜓 ↔ ( ∃ 𝑦 ∈ 𝐶 𝑥 = 𝐴 ∧ 𝜓 ) ) )
7		r19.41v	⊢ ( ∃ 𝑦 ∈ 𝐶 ( 𝑥 = 𝐴 ∧ 𝜓 ) ↔ ( ∃ 𝑦 ∈ 𝐶 𝑥 = 𝐴 ∧ 𝜓 ) )
8	3	pm5.32da	⊢ ( 𝜑 → ( ( 𝑥 = 𝐴 ∧ 𝜓 ) ↔ ( 𝑥 = 𝐴 ∧ 𝜒 ) ) )
9	8	rexbidv	⊢ ( 𝜑 → ( ∃ 𝑦 ∈ 𝐶 ( 𝑥 = 𝐴 ∧ 𝜓 ) ↔ ∃ 𝑦 ∈ 𝐶 ( 𝑥 = 𝐴 ∧ 𝜒 ) ) )
10	7 9	bitr3id	⊢ ( 𝜑 → ( ( ∃ 𝑦 ∈ 𝐶 𝑥 = 𝐴 ∧ 𝜓 ) ↔ ∃ 𝑦 ∈ 𝐶 ( 𝑥 = 𝐴 ∧ 𝜒 ) ) )
11	10	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( ( ∃ 𝑦 ∈ 𝐶 𝑥 = 𝐴 ∧ 𝜓 ) ↔ ∃ 𝑦 ∈ 𝐶 ( 𝑥 = 𝐴 ∧ 𝜒 ) ) )
12	6 11	bitrd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝜓 ↔ ∃ 𝑦 ∈ 𝐶 ( 𝑥 = 𝐴 ∧ 𝜒 ) ) )
13	12	reubidva	⊢ ( 𝜑 → ( ∃! 𝑥 ∈ 𝐵 𝜓 ↔ ∃! 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 ( 𝑥 = 𝐴 ∧ 𝜒 ) ) )
14		reurmo	⊢ ( ∃! 𝑦 ∈ 𝐶 𝑥 = 𝐴 → ∃* 𝑦 ∈ 𝐶 𝑥 = 𝐴 )
15	2 14	syl	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ∃* 𝑦 ∈ 𝐶 𝑥 = 𝐴 )
16	1 15	reuxfrd	⊢ ( 𝜑 → ( ∃! 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐶 ( 𝑥 = 𝐴 ∧ 𝜒 ) ↔ ∃! 𝑦 ∈ 𝐶 𝜒 ) )
17	13 16	bitrd	⊢ ( 𝜑 → ( ∃! 𝑥 ∈ 𝐵 𝜓 ↔ ∃! 𝑦 ∈ 𝐶 𝜒 ) )