MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rabtru Structured version   Visualization version   GIF version

Theorem rabtru 3590
Description: Abstract builder using the constant wff . (Contributed by Thierry Arnoux, 4-May-2020.)
Hypothesis
Ref Expression
rabtru.1 𝑥𝐴
Assertion
Ref Expression
rabtru {𝑥𝐴 ∣ ⊤} = 𝐴

Proof of Theorem rabtru
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 tru 1546 . . 3
2 nfcv 2900 . . . 4 𝑥𝑦
3 rabtru.1 . . . 4 𝑥𝐴
4 nftru 1811 . . . 4 𝑥
5 biidd 265 . . . 4 (𝑥 = 𝑦 → (⊤ ↔ ⊤))
62, 3, 4, 5elrabf 3589 . . 3 (𝑦 ∈ {𝑥𝐴 ∣ ⊤} ↔ (𝑦𝐴 ∧ ⊤))
71, 6mpbiran2 710 . 2 (𝑦 ∈ {𝑥𝐴 ∣ ⊤} ↔ 𝑦𝐴)
87eqriv 2736 1 {𝑥𝐴 ∣ ⊤} = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wtru 1543  wcel 2114  wnfc 2880  {crab 3058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-rab 3063  df-v 3402
This theorem is referenced by:  mptexgf  7008  aciunf1  30588
  Copyright terms: Public domain W3C validator