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Theorem iotaeq 5776
 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 2365 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜑))
2 df-clab 2597 . . . . . . 7 (𝑧 ∈ {𝑥𝜑} ↔ [𝑧 / 𝑥]𝜑)
3 df-clab 2597 . . . . . . 7 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
41, 2, 33bitr4g 302 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → (𝑧 ∈ {𝑥𝜑} ↔ 𝑧 ∈ {𝑦𝜑}))
54eqrdv 2608 . . . . 5 (∀𝑥 𝑥 = 𝑦 → {𝑥𝜑} = {𝑦𝜑})
65eqeq1d 2612 . . . 4 (∀𝑥 𝑥 = 𝑦 → ({𝑥𝜑} = {𝑧} ↔ {𝑦𝜑} = {𝑧}))
76abbidv 2728 . . 3 (∀𝑥 𝑥 = 𝑦 → {𝑧 ∣ {𝑥𝜑} = {𝑧}} = {𝑧 ∣ {𝑦𝜑} = {𝑧}})
87unieqd 4382 . 2 (∀𝑥 𝑥 = 𝑦 {𝑧 ∣ {𝑥𝜑} = {𝑧}} = {𝑧 ∣ {𝑦𝜑} = {𝑧}})
9 df-iota 5768 . 2 (℩𝑥𝜑) = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
10 df-iota 5768 . 2 (℩𝑦𝜑) = {𝑧 ∣ {𝑦𝜑} = {𝑧}}
118, 9, 103eqtr4g 2669 1 (∀𝑥 𝑥 = 𝑦 → (℩𝑥𝜑) = (℩𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473   = wceq 1475  [wsb 1867   ∈ wcel 1977  {cab 2596  {csn 4125  ∪ cuni 4372  ℩cio 5766 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-uni 4373  df-iota 5768 This theorem is referenced by: (None)
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