Theorem bj-rababwv 32061

Description: A weak version of rabab 3196 not using df-clel 2606 nor df-v 3175 (but requiring ax-ext 2590). A version without dv condition is provable by replacing bj-vexwv 32051 with bj-vexw 32049 in the proof, hence requiring ax-13 2234. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-rababwv.1	⊢ 𝜓

Assertion

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

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem bj-rababwv

Step	Hyp	Ref	Expression
1		df-rab 2905	. 2 ⊢ {𝑥 ∈ {𝑦 ∣ 𝜓} ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑)}
2		bj-rababwv.1	. . . . 5 ⊢ 𝜓
3	2	bj-vexwv 32051	. . . 4 ⊢ 𝑥 ∈ {𝑦 ∣ 𝜓}
4	3	biantrur 526	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑))
5	4	bj-abbii 31965	. 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑)}
6	1, 5	eqtr4i 2635	1 ⊢ {𝑥 ∈ {𝑦 ∣ 𝜓} ∣ 𝜑} = {𝑥 ∣ 𝜑}

Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  {crab 2900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-rab 2905

This theorem is referenced by: (None)