Theorem dfiota4 5879

Description: The iota operation using the if operator. (Contributed by Scott Fenton, 6-Oct-2017.) (Proof shortened by JJ, 28-Oct-2021.)

Assertion

Ref	Expression
dfiota4	|- ( iota x ph ) = if ( E! x ph , U. { x | ph } , (/) )

Proof of Theorem dfiota4

Step	Hyp	Ref	Expression
1		iotauni 5863	. 2 |- ( E! x ph -> ( iota x ph ) = U. { x | ph } )
2		iotanul 5866	. 2 |- ( -. E! x ph -> ( iota x ph ) = (/) )
3		ifval 4127	. 2 |- ( ( iota x ph ) = if ( E! x ph , U. { x | ph } , (/) ) <-> ( ( E! x ph -> ( iota x ph ) = U. { x | ph } ) /\ ( -. E! x ph -> ( iota x ph ) = (/) ) ) )
4	1, 2, 3	mpbir2an 955	1 |- ( iota x ph ) = if ( E! x ph , U. { x | ph } , (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   E!weu 2470   {cab 2608   (/)c0 3915   ifcif 4086   U.cuni 4436   iotacio 5849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851

This theorem is referenced by: (None)