Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  iotaval Structured version   Unicode version

Theorem iotaval 5421
 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 5411 . 2
2 vex 2951 . . . . . . 7
3 sbeqalb 3205 . . . . . . . 8
4 equcomi 1691 . . . . . . . 8
53, 4syl6 31 . . . . . . 7
62, 5ax-mp 8 . . . . . 6
76ex 424 . . . . 5
8 equequ2 1698 . . . . . . . . . 10
98equcoms 1693 . . . . . . . . 9
109bibi2d 310 . . . . . . . 8
1110biimpd 199 . . . . . . 7
1211alimdv 1631 . . . . . 6
1312com12 29 . . . . 5
147, 13impbid 184 . . . 4
1514alrimiv 1641 . . 3
16 uniabio 5420 . . 3
1715, 16syl 16 . 2
181, 17syl5eq 2479 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359  wal 1549   wceq 1652   wcel 1725  cab 2421  cvv 2948  cuni 4007  cio 5408 This theorem is referenced by:  iotauni  5422  iota1  5424  iotaex  5427  iota4  5428  iota5  5430  iota5f  25172  iotain  27549  iotaexeu  27550  iotasbc  27551  iotaequ  27561  iotavalb  27562  pm14.24  27564  sbiota1  27566 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-v 2950  df-sbc 3154  df-un 3317  df-sn 3812  df-pr 3813  df-uni 4008  df-iota 5410
 Copyright terms: Public domain W3C validator