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Theorem bj-rabtrALT 33252
Description: Alternate proof of bj-rabtr 33251. (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 3261 . . 3 𝑥{𝑥𝐴 ∣ ⊤}
2 nfcv 2902 . . 3 𝑥𝐴
31, 2cleqf 2928 . 2 ({𝑥𝐴 ∣ ⊤} = 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥𝐴 ∣ ⊤} ↔ 𝑥𝐴))
4 tru 1636 . . 3
5 rabid 3254 . . 3 (𝑥 ∈ {𝑥𝐴 ∣ ⊤} ↔ (𝑥𝐴 ∧ ⊤))
64, 5mpbiran2 992 . 2 (𝑥 ∈ {𝑥𝐴 ∣ ⊤} ↔ 𝑥𝐴)
73, 6mpgbir 1875 1 {𝑥𝐴 ∣ ⊤} = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1632  wtru 1633  wcel 2139  {crab 3054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059
This theorem is referenced by: (None)
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