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Theorem iotaeq 4347
 Description: Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
Assertion
Ref Expression
iotaeq (x x = y → (℩xφ) = (℩yφ))

Proof of Theorem iotaeq
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 drsb1 2022 . . . . . . 7 (x x = y → ([z / x]φ ↔ [z / y]φ))
2 df-clab 2340 . . . . . . 7 (z {x φ} ↔ [z / x]φ)
3 df-clab 2340 . . . . . . 7 (z {y φ} ↔ [z / y]φ)
41, 2, 33bitr4g 279 . . . . . 6 (x x = y → (z {x φ} ↔ z {y φ}))
54eqrdv 2351 . . . . 5 (x x = y → {x φ} = {y φ})
65eqeq1d 2361 . . . 4 (x x = y → ({x φ} = {z} ↔ {y φ} = {z}))
76abbidv 2467 . . 3 (x x = y → {z {x φ} = {z}} = {z {y φ} = {z}})
87unieqd 3902 . 2 (x x = y{z {x φ} = {z}} = {z {y φ} = {z}})
9 df-iota 4339 . 2 (℩xφ) = {z {x φ} = {z}}
10 df-iota 4339 . 2 (℩yφ) = {z {y φ} = {z}}
118, 9, 103eqtr4g 2410 1 (x x = y → (℩xφ) = (℩yφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540   = wceq 1642  [wsb 1648   ∈ wcel 1710  {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: (None)
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