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Theorem ifidss 3541
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 3540 . 2 if(𝜑, 𝐴, 𝐴) ⊆ (𝐴𝐴)
2 unidm 3270 . 2 (𝐴𝐴) = 𝐴
31, 2sseqtri 3181 1 if(𝜑, 𝐴, 𝐴) ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  cun 3119  wss 3121  ifcif 3526
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-if 3527
This theorem is referenced by: (None)
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