Theorem bj-rababw 35075

Description: A weak version of rabab 3461 not using df-clel 2817 nor df-v 3435 (but requiring ax-ext 2710) nor ax-12 2172. (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 3074	. 2 ⊢ {𝑥 ∈ {𝑦 ∣ 𝜓} ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑)}
2		bj-rababw.1	. . . . 5 ⊢ 𝜓
3	2	vexw 2722	. . . 4 ⊢ 𝑥 ∈ {𝑦 ∣ 𝜓}
4	3	biantrur 531	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑))
5	4	abbii 2809	. 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑)}
6	1, 5	eqtr4i 2770	1 ⊢ {𝑥 ∈ {𝑦 ∣ 𝜓} ∣ 𝜑} = {𝑥 ∣ 𝜑}

Syntax hints:   ∧ wa 396   = wceq 1539   ∈ wcel 2107  {cab 2716  {crab 3069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-sb 2069  df-clab 2717  df-cleq 2731  df-rab 3074

This theorem is referenced by: (None)