ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iotabi Structured version   GIF version

Theorem iotabi 4819
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 2148 . . . . . 6 (x(φψ) ↔ {xφ} = {xψ})
21biimpi 113 . . . . 5 (x(φψ) → {xφ} = {xψ})
32eqeq1d 2045 . . . 4 (x(φψ) → ({xφ} = {z} ↔ {xψ} = {z}))
43abbidv 2152 . . 3 (x(φψ) → {z ∣ {xφ} = {z}} = {z ∣ {xψ} = {z}})
54unieqd 3582 . 2 (x(φψ) → {z ∣ {xφ} = {z}} = {z ∣ {xψ} = {z}})
6 df-iota 4810 . 2 (℩xφ) = {z ∣ {xφ} = {z}}
7 df-iota 4810 . 2 (℩xψ) = {z ∣ {xψ} = {z}}
85, 6, 73eqtr4g 2094 1 (x(φψ) → (℩xφ) = (℩xψ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240   = wceq 1242  {cab 2023  {csn 3367   cuni 3571  cio 4808
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-uni 3572  df-iota 4810
This theorem is referenced by:  iotabidv  4831  iotabii  4832  eusvobj1  5442
  Copyright terms: Public domain W3C validator