Theorem bj-rababw 37971

Description: A weak version of rabab 3484 not using df-clel 2839 nor df-v 3456 (but requiring ax-ext 2735) nor ax-12 2216. (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 3417	. 2 ⊢ {𝑥 ∈ {𝑦 ∣ 𝜓} ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑)}
2		bj-rababw.1	. . . . 5 ⊢ 𝜓
3	2	vexw 2747	. . . 4 ⊢ 𝑥 ∈ {𝑦 ∣ 𝜓}
4	3	biantrur 540	. . 3 ⊢ (𝜑 ↔ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑))
5	4	abbii 2830	. 2 ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑦 ∣ 𝜓} ∧ 𝜑)}
6	1, 5	eqtr4i 2789	1 ⊢ {𝑥 ∈ {𝑦 ∣ 𝜓} ∣ 𝜑} = {𝑥 ∣ 𝜑}

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  {crab 3416

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-rab 3417

This theorem is used by: (None)