Theorem euop2 2862

Description: Transfer existential uniqueness to second member of an ordered pair.

Assertion

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

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

Proof of Theorem euop2

Step	Hyp	Ref	Expression
1		opex 2838	. 2 |- <.A, y>. e. V
2		moop2 2857	. 2 |- E*y x = <.A, y>.
3	1, 2	euxfr2 1973	1 |- (E!xE.y(x = <.A, y>. /\ ph) <-> E!yph)

Syntax hints:   <-> wb 153   /\ wa 230   = wceq 997  E.wex 1021  E!weu 1422  <.cop 2463

This theorem is referenced by:  aceq5lem1 4797

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-sep 2758  ax-nul 2765  ax-pow 2798  ax-pr 2835

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-v 1859  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-pw 2454  df-sn 2464  df-pr 2465  df-op 2468

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