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Metamath Proof Explorer

Theorem 2reu5a

Description: Double restricted existential uniqueness in terms of restricted existence and restricted "at most one". (Contributed by Alexander van der Vekens, 17-Jun-2017)

		Ref	Expression
	Assertion	2reu5a	⊢ ( ∃! 𝑥 ∈ 𝐴 ∃! 𝑦 ∈ 𝐵 𝜑 ↔ ( ∃ 𝑥 ∈ 𝐴 ( ∃ 𝑦 ∈ 𝐵 𝜑 ∧ ∃* 𝑦 ∈ 𝐵 𝜑 ) ∧ ∃* 𝑥 ∈ 𝐴 ( ∃ 𝑦 ∈ 𝐵 𝜑 ∧ ∃* 𝑦 ∈ 𝐵 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		reu5	⊢ ( ∃! 𝑥 ∈ 𝐴 ∃! 𝑦 ∈ 𝐵 𝜑 ↔ ( ∃ 𝑥 ∈ 𝐴 ∃! 𝑦 ∈ 𝐵 𝜑 ∧ ∃* 𝑥 ∈ 𝐴 ∃! 𝑦 ∈ 𝐵 𝜑 ) )
2		reu5	⊢ ( ∃! 𝑦 ∈ 𝐵 𝜑 ↔ ( ∃ 𝑦 ∈ 𝐵 𝜑 ∧ ∃* 𝑦 ∈ 𝐵 𝜑 ) )
3	2	rexbii	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃! 𝑦 ∈ 𝐵 𝜑 ↔ ∃ 𝑥 ∈ 𝐴 ( ∃ 𝑦 ∈ 𝐵 𝜑 ∧ ∃* 𝑦 ∈ 𝐵 𝜑 ) )
4	2	rmobii	⊢ ( ∃* 𝑥 ∈ 𝐴 ∃! 𝑦 ∈ 𝐵 𝜑 ↔ ∃* 𝑥 ∈ 𝐴 ( ∃ 𝑦 ∈ 𝐵 𝜑 ∧ ∃* 𝑦 ∈ 𝐵 𝜑 ) )
5	3 4	anbi12i	⊢ ( ( ∃ 𝑥 ∈ 𝐴 ∃! 𝑦 ∈ 𝐵 𝜑 ∧ ∃* 𝑥 ∈ 𝐴 ∃! 𝑦 ∈ 𝐵 𝜑 ) ↔ ( ∃ 𝑥 ∈ 𝐴 ( ∃ 𝑦 ∈ 𝐵 𝜑 ∧ ∃* 𝑦 ∈ 𝐵 𝜑 ) ∧ ∃* 𝑥 ∈ 𝐴 ( ∃ 𝑦 ∈ 𝐵 𝜑 ∧ ∃* 𝑦 ∈ 𝐵 𝜑 ) ) )
6	1 5	bitri	⊢ ( ∃! 𝑥 ∈ 𝐴 ∃! 𝑦 ∈ 𝐵 𝜑 ↔ ( ∃ 𝑥 ∈ 𝐴 ( ∃ 𝑦 ∈ 𝐵 𝜑 ∧ ∃* 𝑦 ∈ 𝐵 𝜑 ) ∧ ∃* 𝑥 ∈ 𝐴 ( ∃ 𝑦 ∈ 𝐵 𝜑 ∧ ∃* 𝑦 ∈ 𝐵 𝜑 ) ) )