Theorem rexcom4b

Description: Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011)

	Ref	Expression
Hypothesis	rexcom4b.1	⊢ 𝐵 ∈ V
Assertion	rexcom4b	⊢ ( ∃ 𝑥 ∃ 𝑦 ∈ 𝐴 ( 𝜑 ∧ 𝑥 = 𝐵 ) ↔ ∃ 𝑦 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		rexcom4b.1	⊢ 𝐵 ∈ V
2		rexcom4a	⊢ ( ∃ 𝑥 ∃ 𝑦 ∈ 𝐴 ( 𝜑 ∧ 𝑥 = 𝐵 ) ↔ ∃ 𝑦 ∈ 𝐴 ( 𝜑 ∧ ∃ 𝑥 𝑥 = 𝐵 ) )
3	1	isseti	⊢ ∃ 𝑥 𝑥 = 𝐵
4	3	biantru	⊢ ( 𝜑 ↔ ( 𝜑 ∧ ∃ 𝑥 𝑥 = 𝐵 ) )
5	4	rexbii	⊢ ( ∃ 𝑦 ∈ 𝐴 𝜑 ↔ ∃ 𝑦 ∈ 𝐴 ( 𝜑 ∧ ∃ 𝑥 𝑥 = 𝐵 ) )
6	2 5	bitr4i	⊢ ( ∃ 𝑥 ∃ 𝑦 ∈ 𝐴 ( 𝜑 ∧ 𝑥 = 𝐵 ) ↔ ∃ 𝑦 ∈ 𝐴 𝜑 )