Theorem 2reu5lem1

Description: Lemma for 2reu5 . Note that E! x e. A E! y e. B ph does not mean "there is exactly one x in A and exactly one y in B such that ph holds"; see comment for 2eu5 . (Contributed by Alexander van der Vekens, 17-Jun-2017)

	Ref	Expression
Assertion	2reu5lem1	⊢ ( ∃! 𝑥 ∈ 𝐴 ∃! 𝑦 ∈ 𝐵 𝜑 ↔ ∃! 𝑥 ∃! 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		df-reu	⊢ ( ∃! 𝑦 ∈ 𝐵 𝜑 ↔ ∃! 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜑 ) )
2	1	reubii	⊢ ( ∃! 𝑥 ∈ 𝐴 ∃! 𝑦 ∈ 𝐵 𝜑 ↔ ∃! 𝑥 ∈ 𝐴 ∃! 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜑 ) )
3		df-reu	⊢ ( ∃! 𝑥 ∈ 𝐴 ∃! 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜑 ) ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ ∃! 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜑 ) ) )
4		euanv	⊢ ( ∃! 𝑦 ( 𝑥 ∈ 𝐴 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝜑 ) ) ↔ ( 𝑥 ∈ 𝐴 ∧ ∃! 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜑 ) ) )
5	4	bicomi	⊢ ( ( 𝑥 ∈ 𝐴 ∧ ∃! 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜑 ) ) ↔ ∃! 𝑦 ( 𝑥 ∈ 𝐴 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝜑 ) ) )
6		3anass	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝜑 ) ) )
7	6	bicomi	⊢ ( ( 𝑥 ∈ 𝐴 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝜑 ) ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑 ) )
8	7	eubii	⊢ ( ∃! 𝑦 ( 𝑥 ∈ 𝐴 ∧ ( 𝑦 ∈ 𝐵 ∧ 𝜑 ) ) ↔ ∃! 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑 ) )
9	5 8	bitri	⊢ ( ( 𝑥 ∈ 𝐴 ∧ ∃! 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜑 ) ) ↔ ∃! 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑 ) )
10	9	eubii	⊢ ( ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ ∃! 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜑 ) ) ↔ ∃! 𝑥 ∃! 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑 ) )
11	3 10	bitri	⊢ ( ∃! 𝑥 ∈ 𝐴 ∃! 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜑 ) ↔ ∃! 𝑥 ∃! 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑 ) )
12	2 11	bitri	⊢ ( ∃! 𝑥 ∈ 𝐴 ∃! 𝑦 ∈ 𝐵 𝜑 ↔ ∃! 𝑥 ∃! 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑 ) )