Theorem elpreqpr 4153

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		elex 3040	. 2 |- ( A e. { B , C } -> A e. _V )
2		elpri 3976	. 2 |- ( A e. { B , C } -> ( A = B \/ A = C ) )
3		elpreqprlem 4152	. . . . 5 |- ( B e. _V -> E. x { B , C } = { B , x } )
4		eleq1 2537	. . . . . 6 |- ( A = B -> ( A e. _V <-> B e. _V ) )
5		preq1 4042	. . . . . . . 8 |- ( A = B -> { A , x } = { B , x } )
6	5	eqeq2d 2481	. . . . . . 7 |- ( A = B -> ( { B , C } = { A , x } <-> { B , C } = { B , x } ) )
7	6	exbidv 1776	. . . . . 6 |- ( A = B -> ( E. x { B , C } = { A , x } <-> E. x { B , C } = { B , x } ) )
8	4, 7	imbi12d 327	. . . . 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 241	. . . 4 |- ( A = B -> ( A e. _V -> E. x { B , C } = { A , x } ) )
10	9	impcom 437	. . 3 |- ( ( A e. _V /\ A = B ) -> E. x { B , C } = { A , x } )
11		elpreqprlem 4152	. . . . . 6 |- ( C e. _V -> E. x { C , B } = { C , x } )
12		prcom 4041	. . . . . . . 8 |- { C , B } = { B , C }
13	12	eqeq1i 2476	. . . . . . 7 |- ( { C , B } = { C , x } <-> { B , C } = { C , x } )
14	13	exbii 1726	. . . . . 6 |- ( E. x { C , B } = { C , x } <-> E. x { B , C } = { C , x } )
15	11, 14	sylib 201	. . . . 5 |- ( C e. _V -> E. x { B , C } = { C , x } )
16		eleq1 2537	. . . . . 6 |- ( A = C -> ( A e. _V <-> C e. _V ) )
17		preq1 4042	. . . . . . . 8 |- ( A = C -> { A , x } = { C , x } )
18	17	eqeq2d 2481	. . . . . . 7 |- ( A = C -> ( { B , C } = { A , x } <-> { B , C } = { C , x } ) )
19	18	exbidv 1776	. . . . . 6 |- ( A = C -> ( E. x { B , C } = { A , x } <-> E. x { B , C } = { C , x } ) )
20	16, 19	imbi12d 327	. . . . 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 241	. . . 4 |- ( A = C -> ( A e. _V -> E. x { B , C } = { A , x } ) )
22	21	impcom 437	. . 3 |- ( ( A e. _V /\ A = C ) -> E. x { B , C } = { A , x } )
23	10, 22	jaodan 802	. 2 |- ( ( A e. _V /\ ( A = B \/ A = C ) ) -> E. x { B , C } = { A , x } )
24	1, 2, 23	syl2anc 673	1 |- ( A e. { B , C } -> E. x { B , C } = { A , x } )

Syntax hints:   -> wi 4   \/ wo 375   = wceq 1452   E.wex 1671   e. wcel 1904   _Vcvv 3031   {cpr 3961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-v 3033  df-dif 3393  df-un 3395  df-nul 3723  df-sn 3960  df-pr 3962

This theorem is referenced by:  elpreqprb  4154