Theorem uniabio 5098

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 2253	. . . . 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 3533	. . . 4 |- { y } = { x | x = y }
4	2, 3	syl6eqr 2190	. . 3 |- ( A. x ( ph <-> x = y ) -> { x | ph } = { y } )
5	4	unieqd 3747	. 2 |- ( A. x ( ph <-> x = y ) -> U. { x | ph } = U. { y } )
6		vex 2689	. . 3 |- y e. _V
7	6	unisn 3752	. 2 |- U. { y } = y
8	5, 7	syl6eq 2188	1 |- ( A. x ( ph <-> x = y ) -> U. { x | ph } = y )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1329   = wceq 1331   {cab 2125   {csn 3527   U.cuni 3736

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-uni 3737

This theorem is referenced by:  iotaval  5099  iotauni  5100