Theorem ifidss 3535

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 3534	. 2 |- if ( ph , A , A ) C_ ( A u. A )
2		unidm 3265	. 2 |- ( A u. A ) = A
3	1, 2	sseqtri 3176	1 |- if ( ph , A , A ) C_ A

Syntax hints:   u. cun 3114   C_ wss 3116   ifcif 3520

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rab 2453  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-if 3521

This theorem is referenced by: (None)