Theorem euxfrdc 3012

Description: Transfer existential uniqueness from a variable x to another variable y contained in expression A. (Contributed by NM, 14-Nov-2004.)

Hypotheses

Ref	Expression
euxfrdc.1	|- A e. _V
euxfrdc.2	|- E! y x = A
euxfrdc.3	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
euxfrdc	|- (DECID E. y E. x ( x = A /\ ps ) -> ( E! x ph <-> E! y ps ) )

Distinct variable groups:   ps, x   ph, y   x, A

Allowed substitution hints:   ph( x)   ps( y)   A( y)

Proof of Theorem euxfrdc

Step	Hyp	Ref	Expression
1		euxfrdc.2	. . . . . 6 |- E! y x = A
2		euex 2116	. . . . . 6 |- ( E! y x = A -> E. y x = A )
3	1, 2	ax-mp 5	. . . . 5 |- E. y x = A
4	3	biantrur 303	. . . 4 |- ( ph <-> ( E. y x = A /\ ph ) )
5		19.41v 1958	. . . 4 |- ( E. y ( x = A /\ ph ) <-> ( E. y x = A /\ ph ) )
6		euxfrdc.3	. . . . . 6 |- ( x = A -> ( ph <-> ps ) )
7	6	pm5.32i 458	. . . . 5 |- ( ( x = A /\ ph ) <-> ( x = A /\ ps ) )
8	7	exbii 1658	. . . 4 |- ( E. y ( x = A /\ ph ) <-> E. y ( x = A /\ ps ) )
9	4, 5, 8	3bitr2i 208	. . 3 |- ( ph <-> E. y ( x = A /\ ps ) )
10	9	eubii 2095	. 2 |- ( E! x ph <-> E! x E. y ( x = A /\ ps ) )
11		euxfrdc.1	. . 3 |- A e. _V
12	1	eumoi 2119	. . 3 |- E* y x = A
13	11, 12	euxfr2dc 3011	. 2 |- (DECID E. y E. x ( x = A /\ ps ) -> ( E! x E. y ( x = A /\ ps ) <-> E! y ps ) )
14	10, 13	bitrid 192	1 |- (DECID E. y E. x ( x = A /\ ps ) -> ( E! x ph <-> E! y ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 846   = wceq 1402   E.wex 1545   E!weu 2086   e. wcel 2209   _Vcvv 2821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-dc 847  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-v 2823

This theorem is referenced by: (None)