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Theorem iotaval 4350
 Description: Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
iotaval
Distinct variable group:   ,
Allowed substitution hints:   (,)

Proof of Theorem iotaval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dfiota2 4340 . 2
2 vex 2862 . . . . . . 7
3 sbeqalb 3098 . . . . . . . 8
4 equcomi 1679 . . . . . . . 8
53, 4syl6 29 . . . . . . 7
62, 5ax-mp 5 . . . . . 6
76ex 423 . . . . 5
8 equequ2 1686 . . . . . . . . . 10
98eqcoms 2356 . . . . . . . . 9
109bibi2d 309 . . . . . . . 8
1110biimpd 198 . . . . . . 7
1211alimdv 1621 . . . . . 6
1312com12 27 . . . . 5
147, 13impbid 183 . . . 4
1514alrimiv 1631 . . 3
16 uniabio 4349 . . 3
1715, 16syl 15 . 2
181, 17syl5eq 2397 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wa 358  wal 1540   wceq 1642   wcel 1710  cab 2339  cvv 2859  cuni 3891  cio 4337 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 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-uni 3892  df-iota 4339 This theorem is referenced by:  iotauni  4351  iota1  4353  iotaex  4356  iota4  4357  iota5  4359
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