Theorem ifidss 3618

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 ( ph , A , A ) C_ A

Proof of Theorem ifidss

Step	Hyp	Ref	Expression
1		ifssun 3617	. 2 |- if ( ph , A , A ) C_ ( A u. A )
2		unidm 3347	. 2 |- ( A u. A ) = A
3	1, 2	sseqtri 3258	1 |- if ( ph , A , A ) C_ A

Syntax hints:   u. cun 3195   C_ wss 3197   ifcif 3602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-if 3603

This theorem is referenced by: (None)