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Theorem bj-ififc 32895
Description: A theorem linking if- and if. (Contributed by BJ, 24-Sep-2019.)
Assertion
Ref Expression
bj-ififc (𝑥 ∈ if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝑥𝐴, 𝑥𝐵))
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝐵

Proof of Theorem bj-ififc
StepHypRef Expression
1 bj-df-ifc 32894 . 2 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥𝐴, 𝑥𝐵)}
21abeq2i 2874 1 (𝑥 ∈ if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝑥𝐴, 𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  if-wif 1050  wcel 2140  ifcif 4231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1051  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-if 4232
This theorem is referenced by: (None)
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