Theorem bj-rababw 36359

Description: A weak version of rabab 3500 not using df-clel 2806 nor df-v 3473 (but requiring ax-ext 2699) nor ax-12 2167. (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 3430	. 2 ⊢ {𝑥 ∈ {𝑦 ∣ 𝜓} ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑)}
2		bj-rababw.1	. . . . 5 ⊢ 𝜓
3	2	vexw 2711	. . . 4 ⊢ 𝑥 ∈ {𝑦 ∣ 𝜓}
4	3	biantrur 530	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑))
5	4	abbii 2798	. 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑)}
6	1, 5	eqtr4i 2759	1 ⊢ {𝑥 ∈ {𝑦 ∣ 𝜓} ∣ 𝜑} = {𝑥 ∣ 𝜑}

Syntax hints:   ∧ wa 395   = wceq 1534   ∈ wcel 2099  {cab 2705  {crab 3429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-rab 3430

This theorem is referenced by: (None)