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Theorem bj-rabtrALT 32595
Description: Alternate proof of bj-rabtr 32594. (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 3111 . . 3 𝑥{𝑥𝐴 ∣ ⊤}
2 nfcv 2761 . . 3 𝑥𝐴
31, 2cleqf 2786 . 2 ({𝑥𝐴 ∣ ⊤} = 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥𝐴 ∣ ⊤} ↔ 𝑥𝐴))
4 tru 1484 . . 3
5 rabid 3106 . . 3 (𝑥 ∈ {𝑥𝐴 ∣ ⊤} ↔ (𝑥𝐴 ∧ ⊤))
64, 5mpbiran2 953 . 2 (𝑥 ∈ {𝑥𝐴 ∣ ⊤} ↔ 𝑥𝐴)
73, 6mpgbir 1723 1 {𝑥𝐴 ∣ ⊤} = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  wtru 1481  wcel 1987  {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-13 2245  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-clel 2617  df-nfc 2750  df-rab 2916
This theorem is referenced by: (None)
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