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Theorem bj-rababw 34595
 Description: A weak version of rabab 3440 not using df-clel 2831 nor df-v 3412 (but requiring ax-ext 2730) nor ax-12 2176. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-rababw.1 𝜓
Assertion
Ref Expression
bj-rababw {𝑥 ∈ {𝑦𝜓} ∣ 𝜑} = {𝑥𝜑}

Proof of Theorem bj-rababw
StepHypRef Expression
1 df-rab 3080 . 2 {𝑥 ∈ {𝑦𝜓} ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑦𝜓} ∧ 𝜑)}
2 bj-rababw.1 . . . . 5 𝜓
32vexw 2742 . . . 4 𝑥 ∈ {𝑦𝜓}
43biantrur 535 . . 3 (𝜑 ↔ (𝑥 ∈ {𝑦𝜓} ∧ 𝜑))
54abbii 2824 . 2 {𝑥𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑦𝜓} ∧ 𝜑)}
61, 5eqtr4i 2785 1 {𝑥 ∈ {𝑦𝜓} ∣ 𝜑} = {𝑥𝜑}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 400   = wceq 1539   ∈ wcel 2112  {cab 2736  {crab 3075 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 1912  ax-6 1971  ax-7 2016  ax-9 2122  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-rab 3080 This theorem is referenced by: (None)
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