Theorem elpreqprlem 4395

Description: Lemma for elpreqpr 4396. (Contributed by Scott Fenton, 7-Dec-2020.) (Revised by AV, 9-Dec-2020.)

Assertion

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

Distinct variable groups:   x, B   x, C

Allowed substitution hint:   V( x)

Proof of Theorem elpreqprlem

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- { B , C } = { B , C }
2		preq2 4269	. . . . . 6 |- ( x = C -> { B , x } = { B , C } )
3	2	eqeq2d 2632	. . . . 5 |- ( x = C -> ( { B , C } = { B , x } <-> { B , C } = { B , C } ) )
4	3	spcegv 3294	. . . 4 |- ( C e. _V -> ( { B , C } = { B , C } -> E. x { B , C } = { B , x } ) )
5	1, 4	mpi 20	. . 3 |- ( C e. _V -> E. x { B , C } = { B , x } )
6	5	a1d 25	. 2 |- ( C e. _V -> ( B e. V -> E. x { B , C } = { B , x } ) )
7		dfsn2 4190	. . . 4 |- { B } = { B , B }
8		preq2 4269	. . . . . 6 |- ( x = B -> { B , x } = { B , B } )
9	8	eqeq2d 2632	. . . . 5 |- ( x = B -> ( { B } = { B , x } <-> { B } = { B , B } ) )
10	9	spcegv 3294	. . . 4 |- ( B e. V -> ( { B } = { B , B } -> E. x { B } = { B , x } ) )
11	7, 10	mpi 20	. . 3 |- ( B e. V -> E. x { B } = { B , x } )
12		prprc2 4301	. . . . 5 |- ( -. C e. _V -> { B , C } = { B } )
13	12	eqeq1d 2624	. . . 4 |- ( -. C e. _V -> ( { B , C } = { B , x } <-> { B } = { B , x } ) )
14	13	exbidv 1850	. . 3 |- ( -. C e. _V -> ( E. x { B , C } = { B , x } <-> E. x { B } = { B , x } ) )
15	11, 14	syl5ibr 236	. 2 |- ( -. C e. _V -> ( B e. V -> E. x { B , C } = { B , x } ) )
16	6, 15	pm2.61i 176	1 |- ( B e. V -> E. x { B , C } = { B , x } )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   {csn 4177   {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:  elpreqpr  4396