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Theorem iotaval 5557
 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 5547 . 2
2 vex 3048 . . . . . . 7
3 sbeqalb 3320 . . . . . . . 8
4 equcomi 1861 . . . . . . . 8
53, 4syl6 34 . . . . . . 7
62, 5ax-mp 5 . . . . . 6
76ex 436 . . . . 5
8 equequ2 1868 . . . . . . . . . 10
98equcoms 1864 . . . . . . . . 9
109bibi2d 320 . . . . . . . 8
1110biimpd 211 . . . . . . 7
1211alimdv 1763 . . . . . 6
1312com12 32 . . . . 5
147, 13impbid 194 . . . 4
1514alrimiv 1773 . . 3
16 uniabio 5556 . . 3
1715, 16syl 17 . 2
181, 17syl5eq 2497 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 188   wa 371  wal 1442   wceq 1444   wcel 1887  cab 2437  cvv 3045  cuni 4198  cio 5544 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-rex 2743  df-v 3047  df-sbc 3268  df-un 3409  df-sn 3969  df-pr 3971  df-uni 4199  df-iota 5546 This theorem is referenced by:  iotauni  5558  iota1  5560  iotaex  5563  iota4  5564  iota5  5566  iota5f  30357  iotain  36768  iotaexeu  36769  iotasbc  36770  iotaequ  36780  iotavalb  36781  pm14.24  36783  sbiota1  36785
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