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

Theorem rabeqcda 3443
Description: When 𝜓 is always true in a context, a restricted class abstraction is equal to the restricting class. Deduction form of rabeqc 3444. (Contributed by Steven Nguyen, 7-Jun-2023.)
Hypothesis
Ref Expression
rabeqcda.1 ((𝜑𝑥𝐴) → 𝜓)
Assertion
Ref Expression
rabeqcda (𝜑 → {𝑥𝐴𝜓} = 𝐴)
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rabeqcda
StepHypRef Expression
1 df-rab 3433 . 2 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
2 rabeqcda.1 . . . . 5 ((𝜑𝑥𝐴) → 𝜓)
32ex 413 . . . 4 (𝜑 → (𝑥𝐴𝜓))
43pm4.71d 562 . . 3 (𝜑 → (𝑥𝐴 ↔ (𝑥𝐴𝜓)))
54eqabdv 2867 . 2 (𝜑𝐴 = {𝑥 ∣ (𝑥𝐴𝜓)})
61, 5eqtr4id 2791 1 (𝜑 → {𝑥𝐴𝜓} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  {cab 2709  {crab 3432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433
This theorem is referenced by:  rabeqc  3444  cmnbascntr  19675  lrold  27399  prjcrv0  41457  mreclat  47700
  Copyright terms: Public domain W3C validator