Theorem reuhyp 3849

Description: A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr 3847.

Hypotheses

Ref	Expression
reuhyp.1	|- (x e. C -> B e. C)
reuhyp.2	|- ((x e. C /\ y e. C) -> (x = A <-> y = B))

Assertion

Ref	Expression
reuhyp	|- (x e. C -> E!y e. C x = A)

Distinct variable groups:   y,B   y,C   x,y

Proof of Theorem reuhyp

Step	Hyp	Ref	Expression
1		equid 1484	. 2 |- x = x
2		reuhyp.1	. . . 4 |- (x e. C -> B e. C)
3	2	adantl 424	. . 3 |- ((x = x /\ x e. C) -> B e. C)
4		reuhyp.2	. . . 4 |- ((x e. C /\ y e. C) -> (x = A <-> y = B))
5	4	3adant1 894	. . 3 |- ((x = x /\ x e. C /\ y e. C) -> (x = A <-> y = B))
6	3, 5	reuhypd 3848	. 2 |- ((x = x /\ x e. C) -> E!y e. C x = A)
7	1, 6	mpan 759	1 |- (x e. C -> E!y e. C x = A)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E!wreu 2107

This theorem is referenced by:  reuunineg 7275  zmax 7433  rebtwnz 7435

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-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-reu 2111  df-v 2294

Copyright terms: Public domain