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Theorem iotabi 4803
Description: Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
Assertion
Ref Expression
iotabi (x(φψ) → (℩xφ) = (℩xψ))

Proof of Theorem iotabi
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 abbi 2133 . . . . . 6 (x(φψ) ↔ {xφ} = {xψ})
21biimpi 113 . . . . 5 (x(φψ) → {xφ} = {xψ})
32eqeq1d 2030 . . . 4 (x(φψ) → ({xφ} = {z} ↔ {xψ} = {z}))
43abbidv 2137 . . 3 (x(φψ) → {z ∣ {xφ} = {z}} = {z ∣ {xψ} = {z}})
54unieqd 3565 . 2 (x(φψ) → {z ∣ {xφ} = {z}} = {z ∣ {xψ} = {z}})
6 df-iota 4794 . 2 (℩xφ) = {z ∣ {xφ} = {z}}
7 df-iota 4794 . 2 (℩xψ) = {z ∣ {xψ} = {z}}
85, 6, 73eqtr4g 2079 1 (x(φψ) → (℩xφ) = (℩xψ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1226   = wceq 1228  {cab 2008  {csn 3350   cuni 3554  cio 4792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-uni 3555  df-iota 4794
This theorem is referenced by:  iotabidv  4815  iotabii  4816  eusvobj1  5423
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