Theorem opeqex 4294

Description: Equivalence of existence implied by equality of ordered pairs. (Contributed by NM, 28-May-2008.)

Assertion

Ref	Expression
opeqex	|- ( <. A , B >. = <. C , D >. -> ( ( A e. _V /\ B e. _V ) <-> ( C e. _V /\ D e. _V ) ) )

Proof of Theorem opeqex

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2269	. . 3 |- ( <. A , B >. = <. C , D >. -> ( x e. <. A , B >. <-> x e. <. C , D >. ) )
2	1	exbidv 1848	. 2 |- ( <. A , B >. = <. C , D >. -> ( E. x x e. <. A , B >. <-> E. x x e. <. C , D >. ) )
3		opm 4278	. 2 |- ( E. x x e. <. A , B >. <-> ( A e. _V /\ B e. _V ) )
4		opm 4278	. 2 |- ( E. x x e. <. C , D >. <-> ( C e. _V /\ D e. _V ) )
5	2, 3, 4	3bitr3g 222	1 |- ( <. A , B >. = <. C , D >. -> ( ( A e. _V /\ B e. _V ) <-> ( C e. _V /\ D e. _V ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   E.wex 1515   e. wcel 2176   _Vcvv 2772   <.cop 3636

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-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642

This theorem is referenced by:  epelg  4337