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Theorem rabeqel 37170
Description: Class element of a restricted class abstraction. (Contributed by Peter Mazsa, 24-Jul-2021.)
Hypotheses
Ref Expression
rabeqel.1 𝐵 = {𝑥𝐴𝜑}
rabeqel.2 (𝑥 = 𝐶 → (𝜑𝜓))
Assertion
Ref Expression
rabeqel (𝐶𝐵 ↔ (𝜓𝐶𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem rabeqel
StepHypRef Expression
1 rabeqel.2 . . 3 (𝑥 = 𝐶 → (𝜑𝜓))
2 rabeqel.1 . . 3 𝐵 = {𝑥𝐴𝜑}
31, 2elrab2 3687 . 2 (𝐶𝐵 ↔ (𝐶𝐴𝜓))
43biancomi 464 1 (𝐶𝐵 ↔ (𝜓𝐶𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  {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-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477
This theorem is referenced by:  elrefrels2  37436  elrefrels3  37437  elcnvrefrels2  37452  elcnvrefrels3  37453  elsymrels2  37471  elsymrels3  37472  elsymrels4  37473  elsymrels5  37474  elrefsymrels2  37487  eltrrels2  37497  eltrrels3  37498  eleqvrels2  37510  eleqvrels3  37511  elfunsALTV  37610  eldisjs  37640
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