Theorem bj-issetw

Description: The closest one can get to isset without using ax-ext . See also vexw . Note that the only disjoint variable condition is between y and A . From there, one can prove isset using eleq2i (which requires ax-ext and df-cleq ). (Contributed by BJ, 29-Apr-2019) (Proof modification is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		bj-issetw.1	⊢ 𝜑
2		bj-issetwt	⊢ ( ∀ 𝑥 𝜑 → ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ ∃ 𝑦 𝑦 = 𝐴 ) )
3	2 1	mpg	⊢ ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ ∃ 𝑦 𝑦 = 𝐴 )