Theorem elpreqpr 4396

Description: Equality and membership rule for pairs. (Contributed by Scott Fenton, 7-Dec-2020.)

Assertion

Ref	Expression
elpreqpr	|- ( A e. { B , C } -> E. x { B , C } = { A , x } )

Distinct variable groups:   x, A   x, B   x, C

Proof of Theorem elpreqpr

Step	Hyp	Ref	Expression
1		elpri 4197	. 2 |- ( A e. { B , C } -> ( A = B \/ A = C ) )
2		elex 3212	. 2 |- ( A e. { B , C } -> A e. _V )
3		elpreqprlem 4395	. . . . 5 |- ( B e. _V -> E. x { B , C } = { B , x } )
4		eleq1 2689	. . . . . 6 |- ( A = B -> ( A e. _V <-> B e. _V ) )
5		preq1 4268	. . . . . . . 8 |- ( A = B -> { A , x } = { B , x } )
6	5	eqeq2d 2632	. . . . . . 7 |- ( A = B -> ( { B , C } = { A , x } <-> { B , C } = { B , x } ) )
7	6	exbidv 1850	. . . . . 6 |- ( A = B -> ( E. x { B , C } = { A , x } <-> E. x { B , C } = { B , x } ) )
8	4, 7	imbi12d 334	. . . . 5 |- ( A = B -> ( ( A e. _V -> E. x { B , C } = { A , x } ) <-> ( B e. _V -> E. x { B , C } = { B , x } ) ) )
9	3, 8	mpbiri 248	. . . 4 |- ( A = B -> ( A e. _V -> E. x { B , C } = { A , x } ) )
10	9	imp 445	. . 3 |- ( ( A = B /\ A e. _V ) -> E. x { B , C } = { A , x } )
11		elpreqprlem 4395	. . . . . 6 |- ( C e. _V -> E. x { C , B } = { C , x } )
12		prcom 4267	. . . . . . . 8 |- { C , B } = { B , C }
13	12	eqeq1i 2627	. . . . . . 7 |- ( { C , B } = { C , x } <-> { B , C } = { C , x } )
14	13	exbii 1774	. . . . . 6 |- ( E. x { C , B } = { C , x } <-> E. x { B , C } = { C , x } )
15	11, 14	sylib 208	. . . . 5 |- ( C e. _V -> E. x { B , C } = { C , x } )
16		eleq1 2689	. . . . . 6 |- ( A = C -> ( A e. _V <-> C e. _V ) )
17		preq1 4268	. . . . . . . 8 |- ( A = C -> { A , x } = { C , x } )
18	17	eqeq2d 2632	. . . . . . 7 |- ( A = C -> ( { B , C } = { A , x } <-> { B , C } = { C , x } ) )
19	18	exbidv 1850	. . . . . 6 |- ( A = C -> ( E. x { B , C } = { A , x } <-> E. x { B , C } = { C , x } ) )
20	16, 19	imbi12d 334	. . . . 5 |- ( A = C -> ( ( A e. _V -> E. x { B , C } = { A , x } ) <-> ( C e. _V -> E. x { B , C } = { C , x } ) ) )
21	15, 20	mpbiri 248	. . . 4 |- ( A = C -> ( A e. _V -> E. x { B , C } = { A , x } ) )
22	21	imp 445	. . 3 |- ( ( A = C /\ A e. _V ) -> E. x { B , C } = { A , x } )
23	10, 22	jaoian 824	. 2 |- ( ( ( A = B \/ A = C ) /\ A e. _V ) -> E. x { B , C } = { A , x } )
24	1, 2, 23	syl2anc 693	1 |- ( A e. { B , C } -> E. x { B , C } = { A , x } )

Syntax hints:   -> wi 4   \/ wo 383   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   {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-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180

This theorem is referenced by:  elpreqprb  4397