Theorem opeqex 4357

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 2296	. . 3 |- ( <. A , B >. = <. C , D >. -> ( x e. <. A , B >. <-> x e. <. C , D >. ) )
2	1	exbidv 1874	. 2 |- ( <. A , B >. = <. C , D >. -> ( E. x x e. <. A , B >. <-> E. x x e. <. C , D >. ) )
3		opm 4341	. 2 |- ( E. x x e. <. A , B >. <-> ( A e. _V /\ B e. _V ) )
4		opm 4341	. 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 1398   E.wex 1541   e. wcel 2203   _Vcvv 2812   <.cop 3685

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4221  ax-pow 4279

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3667  df-sn 3688  df-pr 3689  df-op 3691

This theorem is referenced by:  epelg  4402