New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  brswap2 GIF version

Theorem brswap2 4860
 Description: Binary relationship equivalence for the Swap function. (Contributed by set.mm contributors, 8-Jan-2015.)
Hypotheses
Ref Expression
br1st.1 B V
brswap.2 C V
Assertion
Ref Expression
brswap2 (A Swap B, CA = C, B)

Proof of Theorem brswap2
Dummy variables x y z w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brex 4689 . . 3 (A Swap B, C → (A V B, C V))
21simpld 445 . 2 (A Swap B, CA V)
3 brswap.2 . . . 4 C V
4 br1st.1 . . . 4 B V
53, 4opex 4588 . . 3 C, B V
6 eleq1 2413 . . 3 (A = C, B → (A V ↔ C, B V))
75, 6mpbiri 224 . 2 (A = C, BA V)
84, 3opex 4588 . . 3 B, C V
9 eqeq1 2359 . . . . . 6 (x = A → (x = z, wA = z, w))
109anbi1d 685 . . . . 5 (x = A → ((x = z, w y = w, z) ↔ (A = z, w y = w, z)))
11102exbidv 1628 . . . 4 (x = A → (zw(x = z, w y = w, z) ↔ zw(A = z, w y = w, z)))
12 eqeq1 2359 . . . . . . . . 9 (y = B, C → (y = w, zB, C = w, z))
1312anbi2d 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, zw, z = B, C)
15 opth 4602 . . . . . . . . . . 11 (w, z = B, C ↔ (w = B z = C))
1614, 15bitri 240 . . . . . . . . . 10 (B, C = w, z ↔ (w = B z = C))
1716anbi1i 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))
2017, 18, 193bitr4ri 269 . . . . . . . 8 ((w = B z = C A = z, w) ↔ (A = z, w B, C = w, z))
2113, 20syl6bbr 254 . . . . . . 7 (y = B, C → ((A = z, w y = w, z) ↔ (w = B z = C A = z, w)))
22212exbidv 1628 . . . . . 6 (y = B, C → (zw(A = z, w y = w, z) ↔ zw(w = B z = C A = z, w)))
23 excom 1741 . . . . . 6 (zw(w = B z = C A = z, w) ↔ wz(w = B z = C A = z, w))
2422, 23syl6bb 252 . . . . 5 (y = B, C → (zw(A = z, w y = w, z) ↔ wz(w = B z = C A = z, w)))
25 opeq2 4579 . . . . . . 7 (w = Bz, w = z, B)
2625eqeq2d 2364 . . . . . 6 (w = B → (A = z, wA = z, B))
27 opeq1 4578 . . . . . . 7 (z = Cz, B = C, B)
2827eqeq2d 2364 . . . . . 6 (z = C → (A = z, BA = C, B))
294, 3, 26, 28ceqsex2v 2896 . . . . 5 (wz(w = B z = C A = z, w) ↔ A = C, B)
3024, 29syl6bb 252 . . . 4 (y = B, C → (zw(A = z, w y = w, z) ↔ A = C, B))
31 df-swap 4724 . . . 4 Swap = {x, y zw(x = z, w y = w, z)}
3211, 30, 31brabg 4706 . . 3 ((A V B, C V) → (A Swap B, CA = C, B))
338, 32mpan2 652 . 2 (A V → (A Swap B, CA = C, B))
342, 7, 33pm5.21nii 342 1 (A Swap B, CA = C, B)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859  ⟨cop 4561   class class class wbr 4639   Swap cswap 4718 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-swap 4724 This theorem is referenced by:  dfcnv2  5100  df2nd2  5111  swapf1o  5511  composeex  5820  domfnex  5870  connexex  5913  symex  5916
 Copyright terms: Public domain W3C validator