Theorem iotaint 5229

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 5228	. 2 |- ( E! x ph -> ( iota x ph ) = U. { x | ph } )
2		uniintabim 3908	. 2 |- ( E! x ph -> U. { x | ph } = |^| { x | ph } )
3	1, 2	eqtrd 2226	1 |- ( E! x ph -> ( iota x ph ) = |^| { x | ph } )

Syntax hints:   -> wi 4   = wceq 1364   E!weu 2042   {cab 2179   U.cuni 3836   |^|cint 3871   iotacio 5214

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-sn 3625  df-pr 3626  df-uni 3837  df-int 3872  df-iota 5216

This theorem is referenced by:  bdcriota  15445