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Theorem iotabi 4904
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 2167 . . . . . 6 (∀𝑥(𝜑𝜓) ↔ {𝑥𝜑} = {𝑥𝜓})
21biimpi 117 . . . . 5 (∀𝑥(𝜑𝜓) → {𝑥𝜑} = {𝑥𝜓})
32eqeq1d 2064 . . . 4 (∀𝑥(𝜑𝜓) → ({𝑥𝜑} = {𝑧} ↔ {𝑥𝜓} = {𝑧}))
43abbidv 2171 . . 3 (∀𝑥(𝜑𝜓) → {𝑧 ∣ {𝑥𝜑} = {𝑧}} = {𝑧 ∣ {𝑥𝜓} = {𝑧}})
54unieqd 3619 . 2 (∀𝑥(𝜑𝜓) → {𝑧 ∣ {𝑥𝜑} = {𝑧}} = {𝑧 ∣ {𝑥𝜓} = {𝑧}})
6 df-iota 4895 . 2 (℩𝑥𝜑) = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
7 df-iota 4895 . 2 (℩𝑥𝜓) = {𝑧 ∣ {𝑥𝜓} = {𝑧}}
85, 6, 73eqtr4g 2113 1 (∀𝑥(𝜑𝜓) → (℩𝑥𝜑) = (℩𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257   = wceq 1259  {cab 2042  {csn 3403   cuni 3608  cio 4893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-uni 3609  df-iota 4895
This theorem is referenced by:  iotabidv  4916  iotabii  4917  eusvobj1  5527
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