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Theorem iotabi 6108
 Description: Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
Assertion
Ref Expression
iotabi (∀𝑥(𝜑𝜓) → (℩𝑥𝜑) = (℩𝑥𝜓))

Proof of Theorem iotabi
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 abbi 2902 . . . . . 6 (∀𝑥(𝜑𝜓) ↔ {𝑥𝜑} = {𝑥𝜓})
21biimpi 208 . . . . 5 (∀𝑥(𝜑𝜓) → {𝑥𝜑} = {𝑥𝜓})
32eqeq1d 2780 . . . 4 (∀𝑥(𝜑𝜓) → ({𝑥𝜑} = {𝑧} ↔ {𝑥𝜓} = {𝑧}))
43abbidv 2906 . . 3 (∀𝑥(𝜑𝜓) → {𝑧 ∣ {𝑥𝜑} = {𝑧}} = {𝑧 ∣ {𝑥𝜓} = {𝑧}})
54unieqd 4681 . 2 (∀𝑥(𝜑𝜓) → {𝑧 ∣ {𝑥𝜑} = {𝑧}} = {𝑧 ∣ {𝑥𝜓} = {𝑧}})
6 df-iota 6099 . 2 (℩𝑥𝜑) = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
7 df-iota 6099 . 2 (℩𝑥𝜓) = {𝑧 ∣ {𝑥𝜓} = {𝑧}}
85, 6, 73eqtr4g 2839 1 (∀𝑥(𝜑𝜓) → (℩𝑥𝜑) = (℩𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1599   = wceq 1601  {cab 2763  {csn 4398  ∪ cuni 4671  ℩cio 6097 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-rex 3096  df-uni 4672  df-iota 6099 This theorem is referenced by:  iotabidv  6120  iotabii  6121  eusvobj1  6916  iotasbcq  39597
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