Theorem uniabio 5163

Description: Part of Theorem 8.17 in [Quine] p. 56. This theorem serves as a lemma for the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)

Assertion

Ref	Expression
uniabio	|- ( A. x ( ph <-> x = y ) -> U. { x | ph } = y )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem uniabio

Step	Hyp	Ref	Expression
1		abbi 2280	. . . . 5 |- ( A. x ( ph <-> x = y ) <-> { x | ph } = { x | x = y } )
2	1	biimpi 119	. . . 4 |- ( A. x ( ph <-> x = y ) -> { x | ph } = { x | x = y } )
3		df-sn 3582	. . . 4 |- { y } = { x | x = y }
4	2, 3	eqtr4di 2217	. . 3 |- ( A. x ( ph <-> x = y ) -> { x | ph } = { y } )
5	4	unieqd 3800	. 2 |- ( A. x ( ph <-> x = y ) -> U. { x | ph } = U. { y } )
6		vex 2729	. . 3 |- y e. _V
7	6	unisn 3805	. 2 |- U. { y } = y
8	5, 7	eqtrdi 2215	1 |- ( A. x ( ph <-> x = y ) -> U. { x | ph } = y )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1341   = wceq 1343   {cab 2151   {csn 3576   U.cuni 3789

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-rex 2450  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-uni 3790

This theorem is referenced by:  iotaval  5164  iotauni  5165