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Theorem rabeqcda 3444
Description: When 𝜓 is always true in a context, a restricted class abstraction is equal to the restricting class. Deduction form of rabeqc 3445. (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 3434 . 2 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
2 rabeqcda.1 . . . . 5 ((𝜑𝑥𝐴) → 𝜓)
32ex 414 . . . 4 (𝜑 → (𝑥𝐴𝜓))
43pm4.71d 563 . . 3 (𝜑 → (𝑥𝐴 ↔ (𝑥𝐴𝜓)))
54eqabdv 2868 . 2 (𝜑𝐴 = {𝑥 ∣ (𝑥𝐴𝜓)})
61, 5eqtr4id 2792 1 (𝜑 → {𝑥𝐴𝜓} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {cab 2710  {crab 3433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434
This theorem is referenced by:  rabeqc  3445  cmnbascntr  19673  lrold  27391  prjcrv0  41375  mreclat  47622
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