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Theorem df2nd2 5111
 Description: Alternate definition of the 2nd function. (Contributed by SF, 8-Jan-2015.)
Assertion
Ref Expression
df2nd2 2nd = (1st Swap )

Proof of Theorem df2nd2
Dummy variables x y z w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2862 . . . . . . . 8 y V
21br1st 4858 . . . . . . 7 (w1st yz w = y, z)
32anbi1i 676 . . . . . 6 ((w1st y x Swap w) ↔ (z w = y, z x Swap w))
4 ancom 437 . . . . . 6 ((x Swap w w1st y) ↔ (w1st y x Swap w))
5 19.41v 1901 . . . . . 6 (z(w = y, z x Swap w) ↔ (z w = y, z x Swap w))
63, 4, 53bitr4i 268 . . . . 5 ((x Swap w w1st y) ↔ z(w = y, z x Swap w))
76exbii 1582 . . . 4 (w(x Swap w w1st y) ↔ wz(w = y, z x Swap w))
8 excom 1741 . . . 4 (zw(w = y, z x Swap w) ↔ wz(w = y, z x Swap w))
9 vex 2862 . . . . . . . 8 z V
101, 9opex 4588 . . . . . . 7 y, z V
11 breq2 4643 . . . . . . 7 (w = y, z → (x Swap wx Swap y, z))
1210, 11ceqsexv 2894 . . . . . 6 (w(w = y, z x Swap w) ↔ x Swap y, z)
131, 9brswap2 4860 . . . . . 6 (x Swap y, zx = z, y)
1412, 13bitri 240 . . . . 5 (w(w = y, z x Swap w) ↔ x = z, y)
1514exbii 1582 . . . 4 (zw(w = y, z x Swap w) ↔ z x = z, y)
167, 8, 153bitr2ri 265 . . 3 (z x = z, yw(x Swap w w1st y))
1716opabbii 4626 . 2 {x, y z x = z, y} = {x, y w(x Swap w w1st y)}
18 df-2nd 4797 . 2 2nd = {x, y z x = z, y}
19 df-co 4726 . 2 (1st Swap ) = {x, y w(x Swap w w1st y)}
2017, 18, 193eqtr4i 2383 1 2nd = (1st Swap )
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642  ⟨cop 4561  {copab 4622   class class class wbr 4639  1st c1st 4717   Swap cswap 4718   ∘ ccom 4721  2nd c2nd 4783 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-1st 4723  df-swap 4724  df-co 4726  df-2nd 4797 This theorem is referenced by:  2ndex  5112
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