Theorem preqr2 2482

Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same first element, the second elements are equal.

Hypotheses

Ref	Expression
preqr2.1	|- A e. V
preqr2.2	|- B e. V

Assertion

Ref	Expression
preqr2	|- ({C, A} = {C, B} -> A = B)

Proof of Theorem preqr2

Step	Hyp	Ref	Expression
1		prcom 2447	. . 3 |- {C, A} = {A, C}
2		prcom 2447	. . 3 |- {C, B} = {B, C}
3	1, 2	eqeq12i 1488	. 2 |- ({C, A} = {C, B} <-> {A, C} = {B, C})
4		preqr2.1	. . 3 |- A e. V
5		preqr2.2	. . 3 |- B e. V
6	4, 5	preqr1 2481	. 2 |- ({A, C} = {B, C} -> A = B)
7	3, 6	sylbi 199	1 |- ({C, A} = {C, B} -> A = B)

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958  Vcvv 1811  {cpr 2410

This theorem is referenced by:  preq12b 2483  opth 2787  opprc3 2797  opth2 2800  opthreg 4604

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413

Copyright terms: Public domain