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Theorem iotaeq 4802
 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 1662 . . . . . . 7 (x x = y → ([z / x]φ ↔ [z / y]φ))
2 df-clab 2009 . . . . . . 7 (z {xφ} ↔ [z / x]φ)
3 df-clab 2009 . . . . . . 7 (z {yφ} ↔ [z / y]φ)
41, 2, 33bitr4g 212 . . . . . 6 (x x = y → (z {xφ} ↔ z {yφ}))
54eqrdv 2020 . . . . 5 (x x = y → {xφ} = {yφ})
65eqeq1d 2030 . . . 4 (x x = y → ({xφ} = {z} ↔ {yφ} = {z}))
76abbidv 2137 . . 3 (x x = y → {z ∣ {xφ} = {z}} = {z ∣ {yφ} = {z}})
87unieqd 3565 . 2 (x x = y {z ∣ {xφ} = {z}} = {z ∣ {yφ} = {z}})
9 df-iota 4794 . 2 (℩xφ) = {z ∣ {xφ} = {z}}
10 df-iota 4794 . 2 (℩yφ) = {z ∣ {yφ} = {z}}
118, 9, 103eqtr4g 2079 1 (x x = y → (℩xφ) = (℩yφ))
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1226   = wceq 1228   ∈ wcel 1374  [wsb 1627  {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: (None)
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