Theorem eualeqhb 3824

Description: Even if x is free in A, it is effectively bound when A(x) is single-valued.

Assertion

Ref	Expression
eualeqhb	|- (E!yA.x y = A -> (y = A -> A.x y = A))

Distinct variable groups:   x,y   y,A

Proof of Theorem eualeqhb

Step	Hyp	Ref	Expression
1		visset 2295	. . . . 5 |- y e. _V
2	1	biantrur 794	. . . 4 |- (A.x y = A <-> (y e. _V /\ A.x y = A))
3	2	eubii 1780	. . 3 |- (E!yA.x y = A <-> E!y(y e. _V /\ A.x y = A))
4		df-reu 2111	. . 3 |- (E!y e. _V A.x y = A <-> E!y(y e. _V /\ A.x y = A))
5	3, 4	bitr4i 193	. 2 |- (E!yA.x y = A <-> E!y e. _V A.x y = A)
6		eqeq1 1890	. . . . . 6 |- (y = z -> (y = A <-> z = A))
7	6	albidv 1656	. . . . 5 |- (y = z -> (A.x y = A <-> A.x z = A))
8		ax-17 1317	. . . . . . 7 |- (w e. y -> A.x w e. y)
9		hba1 1350	. . . . . . . . 9 |- (A.x z = A -> A.xA.x z = A)
10		ax-17 1317	. . . . . . . . 9 |- (w e. _V -> A.x w e. _V)
11	9, 10	hbrab 2258	. . . . . . . 8 |- (w e. {z e. _V | A.x z = A} -> A.x w e. {z e. _V | A.x z = A})
12	11	hbuni 3183	. . . . . . 7 |- (w e. U.{z e. _V | A.x z = A} -> A.x w e. U.{z e. _V | A.x z = A})
13	8, 12	hbeq 1995	. . . . . 6 |- (y = U.{z e. _V | A.x z = A} -> A.x y = U.{z e. _V | A.x z = A})
14		eqeq1 1890	. . . . . 6 |- (y = U.{z e. _V | A.x z = A} -> (y = A <-> U.{z e. _V | A.x z = A} = A))
15	13, 14	albid 1459	. . . . 5 |- (y = U.{z e. _V | A.x z = A} -> (A.x y = A <-> A.xU.{z e. _V | A.x z = A} = A))
16	7, 15	reuuni3 3812	. . . 4 |- (E!y e. _V A.x y = A -> A.xU.{z e. _V | A.x z = A} = A)
17	16	19.21bi 1408	. . 3 |- (E!y e. _V A.x y = A -> U.{z e. _V | A.x z = A} = A)
18	15, 16	syl5cbir 228	. . . 4 |- (E!y e. _V A.x y = A -> (y = U.{z e. _V | A.x z = A} -> A.x y = A))
19		eqtr3 1907	. . . 4 |- ((y = A /\ U.{z e. _V | A.x z = A} = A) -> y = U.{z e. _V | A.x z = A})
20	18, 19	syl5 20	. . 3 |- (E!y e. _V A.x y = A -> ((y = A /\ U.{z e. _V | A.x z = A} = A) -> A.x y = A))
21	17, 20	mpan2d 766	. 2 |- (E!y e. _V A.x y = A -> (y = A -> A.x y = A))
22	5, 21	sylbi 216	1 |- (E!yA.x y = A -> (y = A -> A.x y = A))

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E!weu 1771  E!wreu 2107  {crab 2108  _Vcvv 2292  U.cuni 3177

This theorem is referenced by:  eualexeq 3825

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

Copyright terms: Public domain