Theorem prer2 1873

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

Hypotheses

Ref	Expression
prer2.1	⊢ A ∈ V
prer2.2	⊢ B ∈ V

Assertion

Ref	Expression
prer2	⊢ ({C, A} = {C, B} → A = B)

Proof of Theorem prer2

Step	Hyp	Ref	Expression
1		prcom 1840	. . 3 ⊢ {C, A} = {A, C}
2		prcom 1840	. . 3 ⊢ {C, B} = {B, C}
3	1, 2	cleq12i 1114	. 2 ⊢ ({C, A} = {C, B} ↔ {A, C} = {B, C})
4		prer2.1	. . 3 ⊢ A ∈ V
5		prer2.2	. . 3 ⊢ B ∈ V
6	4, 5	preqr1 1872	. 2 ⊢ ({A, C} = {B, C} → A = B)
7	3, 6	sylbi 174	1 ⊢ ({C, A} = {C, B} → A = B)

Syntax hints:   → wi 2   = wceq 1091   ∈ wcel 1092  Vcvv 1348  {cpr 1809

This theorem is referenced by:  preq12b 1874  opth 1898  opprc3 1908  opth2 1909  opthreg 3455

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812

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