Theorem uniabio 5095

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

Proof of Theorem uniabio

Step	Hyp	Ref	Expression
1		eq2ab 2004	. . . . 5 |- ({x | ph} = {x | x = y} <-> A.x(ph <-> x = y))
2	1	biimpri 169	. . . 4 |- (A.x(ph <-> x = y) -> {x | ph} = {x | x = y})
3		df-sn 3049	. . . 4 |- {y} = {x | x = y}
4	2, 3	syl6eqr 1946	. . 3 |- (A.x(ph <-> x = y) -> {x | ph} = {y})
5	4	unieqd 3188	. 2 |- (A.x(ph <-> x = y) -> U.{x | ph} = U.{y})
6		visset 2295	. . 3 |- y e. _V
7	6	unisn 3193	. 2 |- U.{y} = y
8	5, 7	syl6eq 1944	1 |- (A.x(ph <-> x = y) -> U.{x | ph} = y)

Syntax hints:   -> wi 3   <-> wb 163  A.wal 1296   = wceq 1298  {cab 1871  {csn 3044  U.cuni 3177

This theorem is referenced by:  iotaval 5096  iotaequ 5097

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-rex 2110  df-v 2294  df-un 2600  df-sn 3049  df-pr 3050  df-uni 3178

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