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Theorem iotabi 4348
 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 2463 . . . . . 6 (x(φψ) ↔ {x φ} = {x ψ})
21biimpi 186 . . . . 5 (x(φψ) → {x φ} = {x ψ})
32eqeq1d 2361 . . . 4 (x(φψ) → ({x φ} = {z} ↔ {x ψ} = {z}))
43abbidv 2467 . . 3 (x(φψ) → {z {x φ} = {z}} = {z {x ψ} = {z}})
54unieqd 3902 . 2 (x(φψ) → {z {x φ} = {z}} = {z {x ψ} = {z}})
6 df-iota 4339 . 2 (℩xφ) = {z {x φ} = {z}}
7 df-iota 4339 . 2 (℩xψ) = {z {x ψ} = {z}}
85, 6, 73eqtr4g 2410 1 (x(φψ) → (℩xφ) = (℩xψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642  {cab 2339  {csn 3737  ∪cuni 3891  ℩cio 4337 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-uni 3892  df-iota 4339 This theorem is referenced by:  iotabidv  4360  iotabii  4361
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