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Theorem bj-rababw 36863
Description: A weak version of rabab 3509 not using df-clel 2813 nor df-v 3479 (but requiring ax-ext 2705) nor ax-12 2174. (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 3433 . 2 {𝑥 ∈ {𝑦𝜓} ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑦𝜓} ∧ 𝜑)}
2 bj-rababw.1 . . . . 5 𝜓
32vexw 2717 . . . 4 𝑥 ∈ {𝑦𝜓}
43biantrur 530 . . 3 (𝜑 ↔ (𝑥 ∈ {𝑦𝜓} ∧ 𝜑))
54abbii 2806 . 2 {𝑥𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑦𝜓} ∧ 𝜑)}
61, 5eqtr4i 2765 1 {𝑥 ∈ {𝑦𝜓} ∣ 𝜑} = {𝑥𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1536  wcel 2105  {cab 2711  {crab 3432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-rab 3433
This theorem is referenced by: (None)
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