Theorem bj-rexcom4bv

Description: Version of rexcom4b and bj-rexcom4b with a disjoint variable condition on x , V , hence removing dependency on df-sb and df-clab (so that it depends on df-clel and df-rex only on top of first-order logic). Prefer its use over bj-rexcom4b when sufficient (in particular when V is substituted for _V ). Note the V in the hypothesis instead of _V . (Contributed by BJ, 14-Sep-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	bj-rexcom4bv.1	⊢ 𝐵 ∈ 𝑉
Assertion	bj-rexcom4bv	⊢ ( ∃ 𝑥 ∃ 𝑦 ∈ 𝐴 ( 𝜑 ∧ 𝑥 = 𝐵 ) ↔ ∃ 𝑦 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		bj-rexcom4bv.1	⊢ 𝐵 ∈ 𝑉
2		rexcom4a	⊢ ( ∃ 𝑥 ∃ 𝑦 ∈ 𝐴 ( 𝜑 ∧ 𝑥 = 𝐵 ) ↔ ∃ 𝑦 ∈ 𝐴 ( 𝜑 ∧ ∃ 𝑥 𝑥 = 𝐵 ) )
3	1	bj-issetiv	⊢ ∃ 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 )