Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-rababw Structured version   Visualization version   GIF version

Theorem bj-rababw 36416
Description: A weak version of rabab 3492 not using df-clel 2802 nor df-v 3465 (but requiring ax-ext 2696) nor ax-12 2166. (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 3420 . 2 {𝑥 ∈ {𝑦𝜓} ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑦𝜓} ∧ 𝜑)}
2 bj-rababw.1 . . . . 5 𝜓
32vexw 2708 . . . 4 𝑥 ∈ {𝑦𝜓}
43biantrur 529 . . 3 (𝜑 ↔ (𝑥 ∈ {𝑦𝜓} ∧ 𝜑))
54abbii 2795 . 2 {𝑥𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑦𝜓} ∧ 𝜑)}
61, 5eqtr4i 2756 1 {𝑥 ∈ {𝑦𝜓} ∣ 𝜑} = {𝑥𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 394   = wceq 1533  wcel 2098  {cab 2702  {crab 3419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-rab 3420
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator