Theorem bj-rababw

Description: A weak version of rabab not using df-clel nor df-v (but requiring ax-ext ) nor ax-12 . (Contributed by BJ, 16-Jun-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	bj-rababw.1	⊢ 𝜓
Assertion	bj-rababw	⊢ { 𝑥 ∈ { 𝑦 ∣ 𝜓 } ∣ 𝜑 } = { 𝑥 ∣ 𝜑 }

Proof

Step	Hyp	Ref	Expression
1		bj-rababw.1	⊢ 𝜓
2		df-rab	⊢ { 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	abbii	⊢ { 𝑥 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ { 𝑦 ∣ 𝜓 } ∧ 𝜑 ) }
6	2 5	eqtr4i	⊢ { 𝑥 ∈ { 𝑦 ∣ 𝜓 } ∣ 𝜑 } = { 𝑥 ∣ 𝜑 }