Theorem preqr2 2478

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 2443	. . 3 |- {C, A} = {A, C}
2		prcom 2443	. . 3 |- {C, B} = {B, C}
3	1, 2	eqeq12i 1485	. 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 2477	. 2 |- ({A, C} = {B, C} -> A = B)
7	3, 6	sylbi 199	1 |- ({C, A} = {C, B} -> A = B)

Syntax hints:   -> wi 3   = wceq 954   e. wcel 956  Vcvv 1807  {cpr 2406

This theorem is referenced by:  preq12b 2479  opth 2782  opprc3 2792  opth2 2795  opthreg 4584

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-un 2046  df-sn 2408  df-pr 2409

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