Theorem iotaint 5232

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 5231	. 2 |- ( E! x ph -> ( iota x ph ) = U. { x | ph } )
2		uniintabim 3911	. 2 |- ( E! x ph -> U. { x | ph } = |^| { x | ph } )
3	1, 2	eqtrd 2229	1 |- ( E! x ph -> ( iota x ph ) = |^| { x | ph } )

Syntax hints:   -> wi 4   = wceq 1364   E!weu 2045   {cab 2182   U.cuni 3839   |^|cint 3874   iotacio 5217

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-sn 3628  df-pr 3629  df-uni 3840  df-int 3875  df-iota 5219

This theorem is referenced by:  bdcriota  15529