MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iotaeq Structured version   Visualization version   GIF version

Theorem iotaeq 6526
Description: Equality theorem for descriptions. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by Andrew Salmon, 30-Jun-2011.) (New usage is discouraged.)
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 2500 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜑))
2 df-clab 2715 . . . . . . 7 (𝑧 ∈ {𝑥𝜑} ↔ [𝑧 / 𝑥]𝜑)
3 df-clab 2715 . . . . . . 7 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
41, 2, 33bitr4g 314 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → (𝑧 ∈ {𝑥𝜑} ↔ 𝑧 ∈ {𝑦𝜑}))
54eqrdv 2735 . . . . 5 (∀𝑥 𝑥 = 𝑦 → {𝑥𝜑} = {𝑦𝜑})
65eqeq1d 2739 . . . 4 (∀𝑥 𝑥 = 𝑦 → ({𝑥𝜑} = {𝑧} ↔ {𝑦𝜑} = {𝑧}))
76abbidv 2808 . . 3 (∀𝑥 𝑥 = 𝑦 → {𝑧 ∣ {𝑥𝜑} = {𝑧}} = {𝑧 ∣ {𝑦𝜑} = {𝑧}})
87unieqd 4920 . 2 (∀𝑥 𝑥 = 𝑦 {𝑧 ∣ {𝑥𝜑} = {𝑧}} = {𝑧 ∣ {𝑦𝜑} = {𝑧}})
9 df-iota 6514 . 2 (℩𝑥𝜑) = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
10 df-iota 6514 . 2 (℩𝑦𝜑) = {𝑧 ∣ {𝑦𝜑} = {𝑧}}
118, 9, 103eqtr4g 2802 1 (∀𝑥 𝑥 = 𝑦 → (℩𝑥𝜑) = (℩𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1538   = wceq 1540  [wsb 2064  wcel 2108  {cab 2714  {csn 4626   cuni 4907  cio 6512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-13 2377  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-ss 3968  df-uni 4908  df-iota 6514
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator