Theorem brswap2 4861

Description: Binary relationship equivalence for the Swap function. (Contributed by set.mm contributors, 8-Jan-2015.)

Hypotheses

Ref	Expression
br1st.1	|- B e. _V
brswap.2	|- C e. _V

Assertion

Ref	Expression
brswap2	|- (A Swap <.B, C>. <-> A = <.C, B>.)

Proof of Theorem brswap2

Dummy variables x y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brex 4690	. . 3 |- (A Swap <.B, C>. -> (A e. _V /\ <.B, C>. e. _V))
2	1	simpld 445	. 2 |- (A Swap <.B, C>. -> A e. _V)
3		brswap.2	. . . 4 |- C e. _V
4		br1st.1	. . . 4 |- B e. _V
5	3, 4	opex 4589	. . 3 |- <.C, B>. e. _V
6		eleq1 2413	. . 3 |- (A = <.C, B>. -> (A e. _V <-> <.C, B>. e. _V))
7	5, 6	mpbiri 224	. 2 |- (A = <.C, B>. -> A e. _V)
8	4, 3	opex 4589	. . 3 |- <.B, C>. e. _V
9		eqeq1 2359	. . . . . 6 |- (x = A -> (x = <.z, w>. <-> A = <.z, w>.))
10	9	anbi1d 685	. . . . 5 |- (x = A -> ((x = <.z, w>. /\ y = <.w, z>.) <-> (A = <.z, w>. /\ y = <.w, z>.)))
11	10	2exbidv 1628	. . . 4 |- (x = A -> (E.zE.w(x = <.z, w>. /\ y = <.w, z>.) <-> E.zE.w(A = <.z, w>. /\ y = <.w, z>.)))
12		eqeq1 2359	. . . . . . . . 9 |- (y = <.B, C>. -> (y = <.w, z>. <-> <.B, C>. = <.w, z>.))
13	12	anbi2d 684	. . . . . . . 8 |- (y = <.B, C>. -> ((A = <.z, w>. /\ y = <.w, z>.) <-> (A = <.z, w>. /\ <.B, C>. = <.w, z>.)))
14		eqcom 2355	. . . . . . . . . . 11 |- (<.B, C>. = <.w, z>. <-> <.w, z>. = <.B, C>.)
15		opth 4603	. . . . . . . . . . 11 |- (<.w, z>. = <.B, C>. <-> (w = B /\ z = C))
16	14, 15	bitri 240	. . . . . . . . . 10 |- (<.B, C>. = <.w, z>. <-> (w = B /\ z = C))
17	16	anbi1i 676	. . . . . . . . 9 |- ((<.B, C>. = <.w, z>. /\ A = <.z, w>.) <-> ((w = B /\ z = C) /\ A = <.z, w>.))
18		ancom 437	. . . . . . . . 9 |- ((A = <.z, w>. /\ <.B, C>. = <.w, z>.) <-> (<.B, C>. = <.w, z>. /\ A = <.z, w>.))
19		df-3an 936	. . . . . . . . 9 |- ((w = B /\ z = C /\ A = <.z, w>.) <-> ((w = B /\ z = C) /\ A = <.z, w>.))
20	17, 18, 19	3bitr4ri 269	. . . . . . . 8 |- ((w = B /\ z = C /\ A = <.z, w>.) <-> (A = <.z, w>. /\ <.B, C>. = <.w, z>.))
21	13, 20	syl6bbr 254	. . . . . . 7 |- (y = <.B, C>. -> ((A = <.z, w>. /\ y = <.w, z>.) <-> (w = B /\ z = C /\ A = <.z, w>.)))
22	21	2exbidv 1628	. . . . . 6 |- (y = <.B, C>. -> (E.zE.w(A = <.z, w>. /\ y = <.w, z>.) <-> E.zE.w(w = B /\ z = C /\ A = <.z, w>.)))
23		excom 1741	. . . . . 6 |- (E.zE.w(w = B /\ z = C /\ A = <.z, w>.) <-> E.wE.z(w = B /\ z = C /\ A = <.z, w>.))
24	22, 23	syl6bb 252	. . . . 5 |- (y = <.B, C>. -> (E.zE.w(A = <.z, w>. /\ y = <.w, z>.) <-> E.wE.z(w = B /\ z = C /\ A = <.z, w>.)))
25		opeq2 4580	. . . . . . 7 |- (w = B -> <.z, w>. = <.z, B>.)
26	25	eqeq2d 2364	. . . . . 6 |- (w = B -> (A = <.z, w>. <-> A = <.z, B>.))
27		opeq1 4579	. . . . . . 7 |- (z = C -> <.z, B>. = <.C, B>.)
28	27	eqeq2d 2364	. . . . . 6 |- (z = C -> (A = <.z, B>. <-> A = <.C, B>.))
29	4, 3, 26, 28	ceqsex2v 2897	. . . . 5 |- (E.wE.z(w = B /\ z = C /\ A = <.z, w>.) <-> A = <.C, B>.)
30	24, 29	syl6bb 252	. . . 4 |- (y = <.B, C>. -> (E.zE.w(A = <.z, w>. /\ y = <.w, z>.) <-> A = <.C, B>.))
31		df-swap 4725	. . . 4 |- Swap = {<.x, y>. | E.zE.w(x = <.z, w>. /\ y = <.w, z>.)}
32	11, 30, 31	brabg 4707	. . 3 |- ((A e. _V /\ <.B, C>. e. _V) -> (A Swap <.B, C>. <-> A = <.C, B>.))
33	8, 32	mpan2 652	. 2 |- (A e. _V -> (A Swap <.B, C>. <-> A = <.C, B>.))
34	2, 7, 33	pm5.21nii 342	1 |- (A Swap <.B, C>. <-> A = <.C, B>.)

Syntax hints:   <-> wb 176   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2860  <.cop 4562   class class class wbr 4640   Swap cswap 4719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-swap 4725

This theorem is referenced by:  dfcnv2  5101  df2nd2  5112  swapf1o  5512  composeex  5821  domfnex  5871  connexex  5914  symex  5917