Theorem bj-rexvw

Description: A weak version of rexv not using ax-ext (nor df-cleq , df-clel , df-v ), and only core FOL axioms. See also bj-ralvw . (Contributed by BJ, 16-Jun-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	bj-rexvw.1	⊢ 𝜓
Assertion	bj-rexvw	⊢ ( ∃ 𝑥 ∈ { 𝑦 ∣ 𝜓 } 𝜑 ↔ ∃ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		bj-rexvw.1	⊢ 𝜓
2		df-rex	⊢ ( ∃ 𝑥 ∈ { 𝑦 ∣ 𝜓 } 𝜑 ↔ ∃ 𝑥 ( 𝑥 ∈ { 𝑦 ∣ 𝜓 } ∧ 𝜑 ) )
3	1	vexw	⊢ 𝑥 ∈ { 𝑦 ∣ 𝜓 }
4	3	biantrur	⊢ ( 𝜑 ↔ ( 𝑥 ∈ { 𝑦 ∣ 𝜓 } ∧ 𝜑 ) )
5	4	exbii	⊢ ( ∃ 𝑥 𝜑 ↔ ∃ 𝑥 ( 𝑥 ∈ { 𝑦 ∣ 𝜓 } ∧ 𝜑 ) )
6	2 5	bitr4i	⊢ ( ∃ 𝑥 ∈ { 𝑦 ∣ 𝜓 } 𝜑 ↔ ∃ 𝑥 𝜑 )