Theorem euuni2 15663

Description: The unique element such that ph.

Hypothesis

Ref	Expression
euuni2.1	|- (x = A -> (ph <-> ps))

Assertion

Ref	Expression
euuni2	|- ((A e. B /\ E!xph) -> (ps <-> U.{x | ph} = A))

Distinct variable groups:   x,A   ps,x

Proof of Theorem euuni2

Step	Hyp	Ref	Expression
1		euuni2.1	. . . 4 |- (x = A -> (ph <-> ps))
2	1	reuuni2 3811	. . 3 |- ((A e. _V /\ E!x e. _V ph) -> (ps <-> U.{x e. _V | ph} = A))
3		rabab 2308	. . . . 5 |- {x e. _V | ph} = {x | ph}
4	3	unieqi 3187	. . . 4 |- U.{x e. _V | ph} = U.{x | ph}
5	4	eqeq1i 1891	. . 3 |- (U.{x e. _V | ph} = A <-> U.{x | ph} = A)
6	2, 5	syl6bb 595	. 2 |- ((A e. _V /\ E!x e. _V ph) -> (ps <-> U.{x | ph} = A))
7		elisset 2299	. 2 |- (A e. B -> A e. _V)
8		reuv 2307	. . 3 |- (E!x e. _V ph <-> E!xph)
9	8	biimpri 169	. 2 |- (E!xph -> E!x e. _V ph)
10	6, 7, 9	syl2an 503	1 |- ((A e. B /\ E!xph) -> (ps <-> U.{x | ph} = A))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E!weu 1771  {cab 1871  E!wreu 2107  {crab 2108  _Vcvv 2292  U.cuni 3177

This theorem is referenced by:  unirep 15664

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-13 1311  ax-14 1312  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  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-uni 3178

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