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Theorem brswap 5509
 Description: Binary relationship of Swap . (Contributed by SF, 23-Feb-2015.)
Assertion
Ref Expression
brswap (A Swap Bxy(A = x, y B = y, x))
Distinct variable groups:   x,A,y   x,B,y

Proof of Theorem brswap
Dummy variables a b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brex 4689 . 2 (A Swap B → (A V B V))
2 vex 2862 . . . . . 6 x V
3 vex 2862 . . . . . 6 y V
42, 3opex 4588 . . . . 5 x, y V
5 eleq1 2413 . . . . 5 (A = x, y → (A V ↔ x, y V))
64, 5mpbiri 224 . . . 4 (A = x, yA V)
73, 2opex 4588 . . . . 5 y, x V
8 eleq1 2413 . . . . 5 (B = y, x → (B V ↔ y, x V))
97, 8mpbiri 224 . . . 4 (B = y, xB V)
106, 9anim12i 549 . . 3 ((A = x, y B = y, x) → (A V B V))
1110exlimivv 1635 . 2 (xy(A = x, y B = y, x) → (A V B V))
12 eqeq1 2359 . . . . 5 (a = A → (a = x, yA = x, y))
1312anbi1d 685 . . . 4 (a = A → ((a = x, y b = y, x) ↔ (A = x, y b = y, x)))
14132exbidv 1628 . . 3 (a = A → (xy(a = x, y b = y, x) ↔ xy(A = x, y b = y, x)))
15 eqeq1 2359 . . . . 5 (b = B → (b = y, xB = y, x))
1615anbi2d 684 . . . 4 (b = B → ((A = x, y b = y, x) ↔ (A = x, y B = y, x)))
17162exbidv 1628 . . 3 (b = B → (xy(A = x, y b = y, x) ↔ xy(A = x, y B = y, x)))
18 df-swap 4724 . . 3 Swap = {a, b xy(a = x, y b = y, x)}
1914, 17, 18brabg 4706 . 2 ((A V B V) → (A Swap Bxy(A = x, y B = y, x)))
201, 11, 19pm5.21nii 342 1 (A Swap Bxy(A = x, y B = y, x))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃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-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 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:  cnvswap  5510
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