Theorem opeqex 2804

Description: Equivalence of existence implied by equality of ordered pairs.

Assertion

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

Proof of Theorem opeqex

Step	Hyp	Ref	Expression
1		eleq2 1538	. . 3 |- (<.A, B>. = <.C, D>. -> ((/) e. <.A, B>. <-> (/) e. <.C, D>.))
2		opprc1b 2802	. . 3 |- (-. A e. V <-> (/) e. <.A, B>.)
3		opprc1b 2802	. . 3 |- (-. C e. V <-> (/) e. <.C, D>.)
4	1, 2, 3	3bitr4g 557	. 2 |- (<.A, B>. = <.C, D>. -> (-. A e. V <-> -. C e. V))
5	4	con4bid 526	1 |- (<.A, B>. = <.C, D>. -> (A e. V <-> C e. V))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   = wceq 958   e. wcel 960  Vcvv 1814  (/)c0 2283  <.cop 2415

This theorem is referenced by:  oteqex 2805

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-nul 2715

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-nul 2284  df-sn 2416  df-pr 2417  df-op 2420

Copyright terms: Public domain