Theorem preqr1g 4385

Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. Closed form of preqr1 4379. (Contributed by AV, 29-Jan-2021.) (Revised by AV, 18-Sep-2021.)

Assertion

Ref	Expression
preqr1g	|- ( ( A e. V /\ B e. W ) -> ( { A , C } = { B , C } -> A = B ) )

Proof of Theorem preqr1g

Step	Hyp	Ref	Expression
1		simpl 473	. . 3 |- ( ( A e. V /\ B e. W ) -> A e. V )
2		simpr 477	. . 3 |- ( ( A e. V /\ B e. W ) -> B e. W )
3	1, 2	preq1b 4377	. 2 |- ( ( A e. V /\ B e. W ) -> ( { A , C } = { B , C } <-> A = B ) )
4	3	biimpd 219	1 |- ( ( A e. V /\ B e. W ) -> ( { A , C } = { B , C } -> A = B ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cpr 4179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-sn 4178  df-pr 4180

This theorem is referenced by:  umgr2adedgspth  26844