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Theorem iotaeq 6319
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 2528 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜑))
2 df-clab 2797 . . . . . . 7 (𝑧 ∈ {𝑥𝜑} ↔ [𝑧 / 𝑥]𝜑)
3 df-clab 2797 . . . . . . 7 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
41, 2, 33bitr4g 315 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → (𝑧 ∈ {𝑥𝜑} ↔ 𝑧 ∈ {𝑦𝜑}))
54eqrdv 2816 . . . . 5 (∀𝑥 𝑥 = 𝑦 → {𝑥𝜑} = {𝑦𝜑})
65eqeq1d 2820 . . . 4 (∀𝑥 𝑥 = 𝑦 → ({𝑥𝜑} = {𝑧} ↔ {𝑦𝜑} = {𝑧}))
76abbidv 2882 . . 3 (∀𝑥 𝑥 = 𝑦 → {𝑧 ∣ {𝑥𝜑} = {𝑧}} = {𝑧 ∣ {𝑦𝜑} = {𝑧}})
87unieqd 4840 . 2 (∀𝑥 𝑥 = 𝑦 {𝑧 ∣ {𝑥𝜑} = {𝑧}} = {𝑧 ∣ {𝑦𝜑} = {𝑧}})
9 df-iota 6307 . 2 (℩𝑥𝜑) = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
10 df-iota 6307 . 2 (℩𝑦𝜑) = {𝑧 ∣ {𝑦𝜑} = {𝑧}}
118, 9, 103eqtr4g 2878 1 (∀𝑥 𝑥 = 𝑦 → (℩𝑥𝜑) = (℩𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1526   = wceq 1528  [wsb 2060  wcel 2105  {cab 2796  {csn 4557   cuni 4830  cio 6305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2167  ax-13 2381  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-rex 3141  df-uni 4831  df-iota 6307
This theorem is referenced by: (None)
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