Theorem rexcom4

Description: Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004) (Proof shortened by Andrew Salmon, 8-Jun-2011) Reduce axiom dependencies. (Revised by BJ, 13-Jun-2019)

	Ref	Expression
Assertion	rexcom4	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 𝜑 ↔ ∃ 𝑦 ∃ 𝑥 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 𝜑 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ ∃ 𝑦 𝜑 ) )
2		19.42v	⊢ ( ∃ 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ∃ 𝑦 𝜑 ) )
3	2	bicomi	⊢ ( ( 𝑥 ∈ 𝐴 ∧ ∃ 𝑦 𝜑 ) ↔ ∃ 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
4	3	exbii	⊢ ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ ∃ 𝑦 𝜑 ) ↔ ∃ 𝑥 ∃ 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
5		excom	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ∃ 𝑦 ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
6		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
7	6	bicomi	⊢ ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ∃ 𝑥 ∈ 𝐴 𝜑 )
8	7	exbii	⊢ ( ∃ 𝑦 ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ∃ 𝑦 ∃ 𝑥 ∈ 𝐴 𝜑 )
9	5 8	bitri	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ∃ 𝑦 ∃ 𝑥 ∈ 𝐴 𝜑 )
10	4 9	bitri	⊢ ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ ∃ 𝑦 𝜑 ) ↔ ∃ 𝑦 ∃ 𝑥 ∈ 𝐴 𝜑 )
11	1 10	bitri	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 𝜑 ↔ ∃ 𝑦 ∃ 𝑥 ∈ 𝐴 𝜑 )