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Theorem iotabi 5169
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 2284 . . . . . 6 (∀𝑥(𝜑𝜓) ↔ {𝑥𝜑} = {𝑥𝜓})
21biimpi 119 . . . . 5 (∀𝑥(𝜑𝜓) → {𝑥𝜑} = {𝑥𝜓})
32eqeq1d 2179 . . . 4 (∀𝑥(𝜑𝜓) → ({𝑥𝜑} = {𝑧} ↔ {𝑥𝜓} = {𝑧}))
43abbidv 2288 . . 3 (∀𝑥(𝜑𝜓) → {𝑧 ∣ {𝑥𝜑} = {𝑧}} = {𝑧 ∣ {𝑥𝜓} = {𝑧}})
54unieqd 3807 . 2 (∀𝑥(𝜑𝜓) → {𝑧 ∣ {𝑥𝜑} = {𝑧}} = {𝑧 ∣ {𝑥𝜓} = {𝑧}})
6 df-iota 5160 . 2 (℩𝑥𝜑) = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
7 df-iota 5160 . 2 (℩𝑥𝜓) = {𝑧 ∣ {𝑥𝜓} = {𝑧}}
85, 6, 73eqtr4g 2228 1 (∀𝑥(𝜑𝜓) → (℩𝑥𝜑) = (℩𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1346   = wceq 1348  {cab 2156  {csn 3583   cuni 3796  cio 5158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-uni 3797  df-iota 5160
This theorem is referenced by:  iotabidv  5181  iotabii  5182  eusvobj1  5840
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