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

Theorem bj-rabtrALT 36318
Description: Alternate proof of bj-rabtr 36317. (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-rabtrALT {𝑥𝐴 ∣ ⊤} = 𝐴
Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-rabtrALT
StepHypRef Expression
1 nfrab1 3445 . . 3 𝑥{𝑥𝐴 ∣ ⊤}
2 nfcv 2897 . . 3 𝑥𝐴
31, 2cleqf 2928 . 2 ({𝑥𝐴 ∣ ⊤} = 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥𝐴 ∣ ⊤} ↔ 𝑥𝐴))
4 tru 1537 . . 3
5 rabid 3446 . . 3 (𝑥 ∈ {𝑥𝐴 ∣ ⊤} ↔ (𝑥𝐴 ∧ ⊤))
64, 5mpbiran2 707 . 2 (𝑥 ∈ {𝑥𝐴 ∣ ⊤} ↔ 𝑥𝐴)
73, 6mpgbir 1793 1 {𝑥𝐴 ∣ ⊤} = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  wtru 1534  wcel 2098  {crab 3426
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-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-rab 3427
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator