Theorem uniabio 4877

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 2151	. . . . 5 |- ( A. x ( ph <-> x = y ) <-> { x | ph } = { x | x = y } )
2	1	biimpi 113	. . . 4 |- ( A. x ( ph <-> x = y ) -> { x | ph } = { x | x = y } )
3		df-sn 3381	. . . 4 |- { y } = { x | x = y }
4	2, 3	syl6eqr 2090	. . 3 |- ( A. x ( ph <-> x = y ) -> { x | ph } = { y } )
5	4	unieqd 3591	. 2 |- ( A. x ( ph <-> x = y ) -> U. { x | ph } = U. { y } )
6		vex 2560	. . 3 |- y e. _V
7	6	unisn 3596	. 2 |- U. { y } = y
8	5, 7	syl6eq 2088	1 |- ( A. x ( ph <-> x = y ) -> U. { x | ph } = y )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   = wceq 1243   {cab 2026   {csn 3375   U.cuni 3580

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-uni 3581

This theorem is referenced by:  iotaval  4878  iotauni  4879