Theorem euop2 4974

Description: Transfer existential uniqueness to second member of an ordered pair. (Contributed by NM, 10-Apr-2004.)

Hypothesis

Ref	Expression
euop2.1	|- A e. _V

Assertion

Ref	Expression
euop2	|- ( E! x E. y ( x = <. A , y >. /\ ph ) <-> E! y ph )

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

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

Proof of Theorem euop2

Step	Hyp	Ref	Expression
1		opex 4932	. 2 |- <. A , y >. e. _V
2		euop2.1	. . 3 |- A e. _V
3	2	moop2 4966	. 2 |- E* y x = <. A , y >.
4	1, 3	euxfr2 3391	1 |- ( E! x E. y ( x = <. A , y >. /\ ph ) <-> E! y ph )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   _Vcvv 3200   <.cop 4183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184

This theorem is referenced by:  dfac5lem1  8946