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Theorem iotabi 6003
 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 2885 . . . . . 6 (∀𝑥(𝜑𝜓) ↔ {𝑥𝜑} = {𝑥𝜓})
21biimpi 206 . . . . 5 (∀𝑥(𝜑𝜓) → {𝑥𝜑} = {𝑥𝜓})
32eqeq1d 2772 . . . 4 (∀𝑥(𝜑𝜓) → ({𝑥𝜑} = {𝑧} ↔ {𝑥𝜓} = {𝑧}))
43abbidv 2889 . . 3 (∀𝑥(𝜑𝜓) → {𝑧 ∣ {𝑥𝜑} = {𝑧}} = {𝑧 ∣ {𝑥𝜓} = {𝑧}})
54unieqd 4582 . 2 (∀𝑥(𝜑𝜓) → {𝑧 ∣ {𝑥𝜑} = {𝑧}} = {𝑧 ∣ {𝑥𝜓} = {𝑧}})
6 df-iota 5994 . 2 (℩𝑥𝜑) = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
7 df-iota 5994 . 2 (℩𝑥𝜓) = {𝑧 ∣ {𝑥𝜓} = {𝑧}}
85, 6, 73eqtr4g 2829 1 (∀𝑥(𝜑𝜓) → (℩𝑥𝜑) = (℩𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1628   = wceq 1630  {cab 2756  {csn 4314  ∪ cuni 4572  ℩cio 5992 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-ext 2750 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-rex 3066  df-uni 4573  df-iota 5994 This theorem is referenced by:  iotabidv  6015  iotabii  6016  eusvobj1  6786  iotasbcq  39157
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