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Theorem iotabi 5824
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 2734 . . . . . 6 (∀𝑥(𝜑𝜓) ↔ {𝑥𝜑} = {𝑥𝜓})
21biimpi 206 . . . . 5 (∀𝑥(𝜑𝜓) → {𝑥𝜑} = {𝑥𝜓})
32eqeq1d 2623 . . . 4 (∀𝑥(𝜑𝜓) → ({𝑥𝜑} = {𝑧} ↔ {𝑥𝜓} = {𝑧}))
43abbidv 2738 . . 3 (∀𝑥(𝜑𝜓) → {𝑧 ∣ {𝑥𝜑} = {𝑧}} = {𝑧 ∣ {𝑥𝜓} = {𝑧}})
54unieqd 4417 . 2 (∀𝑥(𝜑𝜓) → {𝑧 ∣ {𝑥𝜑} = {𝑧}} = {𝑧 ∣ {𝑥𝜓} = {𝑧}})
6 df-iota 5815 . 2 (℩𝑥𝜑) = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
7 df-iota 5815 . 2 (℩𝑥𝜓) = {𝑧 ∣ {𝑥𝜓} = {𝑧}}
85, 6, 73eqtr4g 2680 1 (∀𝑥(𝜑𝜓) → (℩𝑥𝜑) = (℩𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478   = wceq 1480  {cab 2607  {csn 4153   cuni 4407  cio 5813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-uni 4408  df-iota 5815
This theorem is referenced by:  iotabidv  5836  iotabii  5837  eusvobj1  6604  iotasbcq  38147
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