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Theorem rabeqel 34518
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 3561 . 2 (𝐶𝐵 ↔ (𝐶𝐴𝜓))
43biancomi 455 1 (𝐶𝐵 ↔ (𝜓𝐶𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  {crab 3094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2778
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-rab 3099  df-v 3388
This theorem is referenced by:  elrefrels2  34760  elrefrels3  34761  elcnvrefrels2  34773  elcnvrefrels3  34774  elsymrels2  34792  elsymrels3  34793  elsymrels4  34794  elsymrels5  34795  elrefsymrels2  34808  eltrrels2  34818  eltrrels3  34819  eleqvrels2  34829  eleqvrels3  34830
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