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Theorem iotaeq 4818
Description: Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
Assertion
Ref Expression
iotaeq (x x = y → (℩xφ) = (℩yφ))

Proof of Theorem iotaeq
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 drsb1 1677 . . . . . . 7 (x x = y → ([z / x]φ ↔ [z / y]φ))
2 df-clab 2024 . . . . . . 7 (z {xφ} ↔ [z / x]φ)
3 df-clab 2024 . . . . . . 7 (z {yφ} ↔ [z / y]φ)
41, 2, 33bitr4g 212 . . . . . 6 (x x = y → (z {xφ} ↔ z {yφ}))
54eqrdv 2035 . . . . 5 (x x = y → {xφ} = {yφ})
65eqeq1d 2045 . . . 4 (x x = y → ({xφ} = {z} ↔ {yφ} = {z}))
76abbidv 2152 . . 3 (x x = y → {z ∣ {xφ} = {z}} = {z ∣ {yφ} = {z}})
87unieqd 3582 . 2 (x x = y {z ∣ {xφ} = {z}} = {z ∣ {yφ} = {z}})
9 df-iota 4810 . 2 (℩xφ) = {z ∣ {xφ} = {z}}
10 df-iota 4810 . 2 (℩yφ) = {z ∣ {yφ} = {z}}
118, 9, 103eqtr4g 2094 1 (x x = y → (℩xφ) = (℩yφ))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1240   = wceq 1242   wcel 1390  [wsb 1642  {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: (None)
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