Theorem uniabio 5861

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 2737	. . . . 5 |- ( A. x ( ph <-> x = y ) <-> { x | ph } = { x | x = y } )
2	1	biimpi 206	. . . 4 |- ( A. x ( ph <-> x = y ) -> { x | ph } = { x | x = y } )
3		df-sn 4178	. . . 4 |- { y } = { x | x = y }
4	2, 3	syl6eqr 2674	. . 3 |- ( A. x ( ph <-> x = y ) -> { x | ph } = { y } )
5	4	unieqd 4446	. 2 |- ( A. x ( ph <-> x = y ) -> U. { x | ph } = U. { y } )
6		vex 3203	. . 3 |- y e. _V
7	6	unisn 4451	. 2 |- U. { y } = y
8	5, 7	syl6eq 2672	1 |- ( A. x ( ph <-> x = y ) -> U. { x | ph } = y )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   {cab 2608   {csn 4177   U.cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-un 3579  df-sn 4178  df-pr 4180  df-uni 4437

This theorem is referenced by:  iotaval  5862  iotauni  5863