Theorem bj-rexcom4b

Description: Remove from rexcom4b dependency on ax-ext and ax-13 (and on df-or , df-cleq , df-nfc , df-v ). The hypothesis uses V instead of _V (see bj-isseti for the motivation). Use bj-rexcom4bv instead when sufficient (in particular when V is substituted for _V ). (Contributed by BJ, 16-Jun-2019) (Proof modification is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		bj-rexcom4b.1	⊢ 𝐵 ∈ 𝑉
2		rexcom4a	⊢ ( ∃ 𝑥 ∃ 𝑦 ∈ 𝐴 ( 𝜑 ∧ 𝑥 = 𝐵 ) ↔ ∃ 𝑦 ∈ 𝐴 ( 𝜑 ∧ ∃ 𝑥 𝑥 = 𝐵 ) )
3	1	bj-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 )