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

Theorem bj-rababwv 32541
Description: A weak version of rabab 3212 not using df-clel 2617 nor df-v 3191 (but requiring ax-ext 2601). A version without dv condition is provable by replacing bj-vexwv 32531 with bj-vexw 32529 in the proof, hence requiring ax-13 2245. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-rababwv.1 𝜓
Assertion
Ref Expression
bj-rababwv {𝑥 ∈ {𝑦𝜓} ∣ 𝜑} = {𝑥𝜑}
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem bj-rababwv
StepHypRef Expression
1 df-rab 2916 . 2 {𝑥 ∈ {𝑦𝜓} ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑦𝜓} ∧ 𝜑)}
2 bj-rababwv.1 . . . . 5 𝜓
32bj-vexwv 32531 . . . 4 𝑥 ∈ {𝑦𝜓}
43biantrur 527 . . 3 (𝜑 ↔ (𝑥 ∈ {𝑦𝜓} ∧ 𝜑))
54bj-abbii 32447 . 2 {𝑥𝜑} = {𝑥 ∣ (𝑥 ∈ {𝑦𝜓} ∧ 𝜑)}
61, 5eqtr4i 2646 1 {𝑥 ∈ {𝑦𝜓} ∣ 𝜑} = {𝑥𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wcel 1987  {cab 2607  {crab 2911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-rab 2916
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator