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Theorem elswap 4740
Description: Membership in the Swap function. (Contributed by SF, 6-Jan-2015.)
Assertion
Ref Expression
elswap Swap
Distinct variable group:   ,,

Proof of Theorem elswap
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-swap 4724 . . . 4 Swap
21eleq2i 2417 . . 3 Swap
3 elopab 4696 . . 3
42, 3bitri 240 . 2 Swap
5 exrot4 1745 . . 3
6 19.42vv 1907 . . . 4
762exbii 1583 . . 3
8 df-3an 936 . . . . . . 7
9 ancom 437 . . . . . . 7
108, 9bitr2i 241 . . . . . 6
11102exbii 1583 . . . . 5
12 vex 2862 . . . . . . 7
13 vex 2862 . . . . . . 7
1412, 13opex 4588 . . . . . 6
1513, 12opex 4588 . . . . . 6
16 opeq1 4578 . . . . . . 7
1716eqeq2d 2364 . . . . . 6
18 opeq2 4579 . . . . . . 7
1918eqeq2d 2364 . . . . . 6
2014, 15, 17, 19ceqsex2v 2896 . . . . 5
2111, 20bitri 240 . . . 4
22212exbii 1583 . . 3
235, 7, 223bitr3i 266 . 2
244, 23bitri 240 1 Swap
Colors of variables: wff setvar class
Syntax hints:   wb 176   wa 358   w3a 934  wex 1541   wceq 1642   wcel 1710  cop 4561  copab 4622   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-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
This theorem is referenced by:  dfswap2  4741
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