Theorem bj-rababw 36876

Description: A weak version of rabab 3481 not using df-clel 2804 nor df-v 3452 (but requiring ax-ext 2702) nor ax-12 2178. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-rababw.1	⊢ 𝜓

Assertion

Ref	Expression
bj-rababw	⊢ {𝑥 ∈ {𝑦 ∣ 𝜓} ∣ 𝜑} = {𝑥 ∣ 𝜑}

Proof of Theorem bj-rababw

Step	Hyp	Ref	Expression
1		df-rab 3409	. 2 ⊢ {𝑥 ∈ {𝑦 ∣ 𝜓} ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑)}
2		bj-rababw.1	. . . . 5 ⊢ 𝜓
3	2	vexw 2714	. . . 4 ⊢ 𝑥 ∈ {𝑦 ∣ 𝜓}
4	3	biantrur 530	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑))
5	4	abbii 2797	. 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑)}
6	1, 5	eqtr4i 2756	1 ⊢ {𝑥 ∈ {𝑦 ∣ 𝜓} ∣ 𝜑} = {𝑥 ∣ 𝜑}

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2708  {crab 3408

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-rab 3409

This theorem is referenced by: (None)