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Theorem iotaeq 6082
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 2538 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜑))
2 df-clab 2804 . . . . . . 7 (𝑧 ∈ {𝑥𝜑} ↔ [𝑧 / 𝑥]𝜑)
3 df-clab 2804 . . . . . . 7 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
41, 2, 33bitr4g 305 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → (𝑧 ∈ {𝑥𝜑} ↔ 𝑧 ∈ {𝑦𝜑}))
54eqrdv 2815 . . . . 5 (∀𝑥 𝑥 = 𝑦 → {𝑥𝜑} = {𝑦𝜑})
65eqeq1d 2819 . . . 4 (∀𝑥 𝑥 = 𝑦 → ({𝑥𝜑} = {𝑧} ↔ {𝑦𝜑} = {𝑧}))
76abbidv 2936 . . 3 (∀𝑥 𝑥 = 𝑦 → {𝑧 ∣ {𝑥𝜑} = {𝑧}} = {𝑧 ∣ {𝑦𝜑} = {𝑧}})
87unieqd 4651 . 2 (∀𝑥 𝑥 = 𝑦 {𝑧 ∣ {𝑥𝜑} = {𝑧}} = {𝑧 ∣ {𝑦𝜑} = {𝑧}})
9 df-iota 6074 . 2 (℩𝑥𝜑) = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
10 df-iota 6074 . 2 (℩𝑦𝜑) = {𝑧 ∣ {𝑦𝜑} = {𝑧}}
118, 9, 103eqtr4g 2876 1 (∀𝑥 𝑥 = 𝑦 → (℩𝑥𝜑) = (℩𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1635   = wceq 1637  [wsb 2061  wcel 2157  {cab 2803  {csn 4381   cuni 4641  cio 6072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-rex 3113  df-uni 4642  df-iota 6074
This theorem is referenced by: (None)
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