Theorem iotaint 4993

Description: Equivalence between two different forms of iota. (Contributed by Mario Carneiro, 24-Dec-2016.)

Assertion

Ref	Expression
iotaint	|- ( E! x ph -> ( iota x ph ) = |^| { x | ph } )

Proof of Theorem iotaint

Step	Hyp	Ref	Expression
1		iotauni 4992	. 2 |- ( E! x ph -> ( iota x ph ) = U. { x | ph } )
2		uniintabim 3725	. 2 |- ( E! x ph -> U. { x | ph } = |^| { x | ph } )
3	1, 2	eqtrd 2120	1 |- ( E! x ph -> ( iota x ph ) = |^| { x | ph } )

Syntax hints:   -> wi 4   = wceq 1289   E!weu 1948   {cab 2074   U.cuni 3653   |^|cint 3688   iotacio 4978

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-sn 3452  df-pr 3453  df-uni 3654  df-int 3689  df-iota 4980

This theorem is referenced by:  bdcriota  11774