Theorem preq2b 4378

Description: Biconditional equality lemma for unordered pairs, deduction form. Two unordered pairs have the same first element iff the second elements are equal. (Contributed by AV, 18-Dec-2020.)

Hypotheses

Ref	Expression
preq1b.a	|- ( ph -> A e. V )
preq1b.b	|- ( ph -> B e. W )

Assertion

Ref	Expression
preq2b	|- ( ph -> ( { C , A } = { C , B } <-> A = B ) )

Proof of Theorem preq2b

Step	Hyp	Ref	Expression
1		prcom 4267	. . 3 |- { C , A } = { A , C }
2		prcom 4267	. . 3 |- { C , B } = { B , C }
3	1, 2	eqeq12i 2636	. 2 |- ( { C , A } = { C , B } <-> { A , C } = { B , C } )
4		preq1b.a	. . 3 |- ( ph -> A e. V )
5		preq1b.b	. . 3 |- ( ph -> B e. W )
6	4, 5	preq1b 4377	. 2 |- ( ph -> ( { A , C } = { B , C } <-> A = B ) )
7	3, 6	syl5bb 272	1 |- ( ph -> ( { C , A } = { C , B } <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 196   = 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:  umgr2v2enb1  26422  clsk1indlem4  38342