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Theorem ifidss 3521
Description: A conditional class whose two alternatives are equal is included in that alternative. With excluded middle, we can prove it is equal to it. (Contributed by BJ, 15-Aug-2024.)
Assertion
Ref Expression
ifidss if(𝜑, 𝐴, 𝐴) ⊆ 𝐴

Proof of Theorem ifidss
StepHypRef Expression
1 ifssun 3520 . 2 if(𝜑, 𝐴, 𝐴) ⊆ (𝐴𝐴)
2 unidm 3251 . 2 (𝐴𝐴) = 𝐴
31, 2sseqtri 3162 1 if(𝜑, 𝐴, 𝐴) ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  cun 3100  wss 3102  ifcif 3506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rab 2444  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-if 3507
This theorem is referenced by: (None)
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