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Theorem dfswap3 5728
 Description: Alternate definition of Swap as an operator abstraction. (Contributed by SF, 23-Feb-2015.)
Assertion
Ref Expression
dfswap3 Swap = {x, y, z z = y, x}
Distinct variable group:   x,y,z

Proof of Theorem dfswap3
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 df-swap 4724 . 2 Swap = {w, z xy(w = x, y z = y, x)}
2 dfoprab2 5558 . 2 {x, y, z z = y, x} = {w, z xy(w = x, y z = y, x)}
31, 2eqtr4i 2376 1 Swap = {x, y, z z = y, x}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642  ⟨cop 4561  {copab 4622   Swap cswap 4718  {coprab 5527 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-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-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  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-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-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-opab 4623  df-swap 4724  df-oprab 5528 This theorem is referenced by:  dfswap4  5729
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