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Theorem bj-rabtrALT 33420
Description: Alternate proof of bj-rabtr 33419. (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 3305 . . 3 𝑥{𝑥𝐴 ∣ ⊤}
2 nfcv 2942 . . 3 𝑥𝐴
31, 2cleqf 2968 . 2 ({𝑥𝐴 ∣ ⊤} = 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥𝐴 ∣ ⊤} ↔ 𝑥𝐴))
4 tru 1658 . . 3
5 rabid 3298 . . 3 (𝑥 ∈ {𝑥𝐴 ∣ ⊤} ↔ (𝑥𝐴 ∧ ⊤))
64, 5mpbiran2 702 . 2 (𝑥 ∈ {𝑥𝐴 ∣ ⊤} ↔ 𝑥𝐴)
73, 6mpgbir 1895 1 {𝑥𝐴 ∣ ⊤} = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1653  wtru 1654  wcel 2157  {crab 3094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-rab 3099
This theorem is referenced by: (None)
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