Theorem preqr2 4126

Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same first element, the second elements are equal. (Contributed by NM, 5-Aug-1993.)

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 3799	. . 3 |- {C, A} = {A, C}
2		prcom 3799	. . 3 |- {C, B} = {B, C}
3	1, 2	eqeq12i 2366	. 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 4125	. 2 |- ({A, C} = {B, C} -> A = B)
7	3, 6	sylbi 187	1 |- ({C, A} = {C, B} -> A = B)

Syntax hints:   -> wi 4   = wceq 1642   e. wcel 1710  _Vcvv 2860  {cpr 3739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743

This theorem is referenced by:  preqr2g  4127  preq12b  4128  opkthg  4132