Theorem uniabio 5170

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 2284	. . . . 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 3589	. . . 4 |- { y } = { x | x = y }
4	2, 3	eqtr4di 2221	. . 3 |- ( A. x ( ph <-> x = y ) -> { x | ph } = { y } )
5	4	unieqd 3807	. 2 |- ( A. x ( ph <-> x = y ) -> U. { x | ph } = U. { y } )
6		vex 2733	. . 3 |- y e. _V
7	6	unisn 3812	. 2 |- U. { y } = y
8	5, 7	eqtrdi 2219	1 |- ( A. x ( ph <-> x = y ) -> U. { x | ph } = y )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1346   = wceq 1348   {cab 2156   {csn 3583   U.cuni 3796

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-uni 3797

This theorem is referenced by:  iotaval  5171  iotauni  5172