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Theorem iotaeq 4975
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 1727 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜑))
2 df-clab 2075 . . . . . . 7 (𝑧 ∈ {𝑥𝜑} ↔ [𝑧 / 𝑥]𝜑)
3 df-clab 2075 . . . . . . 7 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
41, 2, 33bitr4g 221 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → (𝑧 ∈ {𝑥𝜑} ↔ 𝑧 ∈ {𝑦𝜑}))
54eqrdv 2086 . . . . 5 (∀𝑥 𝑥 = 𝑦 → {𝑥𝜑} = {𝑦𝜑})
65eqeq1d 2096 . . . 4 (∀𝑥 𝑥 = 𝑦 → ({𝑥𝜑} = {𝑧} ↔ {𝑦𝜑} = {𝑧}))
76abbidv 2205 . . 3 (∀𝑥 𝑥 = 𝑦 → {𝑧 ∣ {𝑥𝜑} = {𝑧}} = {𝑧 ∣ {𝑦𝜑} = {𝑧}})
87unieqd 3659 . 2 (∀𝑥 𝑥 = 𝑦 {𝑧 ∣ {𝑥𝜑} = {𝑧}} = {𝑧 ∣ {𝑦𝜑} = {𝑧}})
9 df-iota 4967 . 2 (℩𝑥𝜑) = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
10 df-iota 4967 . 2 (℩𝑦𝜑) = {𝑧 ∣ {𝑦𝜑} = {𝑧}}
118, 9, 103eqtr4g 2145 1 (∀𝑥 𝑥 = 𝑦 → (℩𝑥𝜑) = (℩𝑦𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1287   = wceq 1289  wcel 1438  [wsb 1692  {cab 2074  {csn 3441   cuni 3648  cio 4965
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-uni 3649  df-iota 4967
This theorem is referenced by: (None)
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