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Theorem iotaval 6515
Description: Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.) Remove dependency on ax-10 2138, ax-11 2155, ax-12 2172. (Revised by SN, 23-Nov-2024.)
Assertion
Ref Expression
iotaval (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem iotaval
StepHypRef Expression
1 abbi 2801 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑥𝑥 = 𝑦})
2 df-sn 4630 . . 3 {𝑦} = {𝑥𝑥 = 𝑦}
31, 2eqtr4di 2791 . 2 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑦})
4 iotaval2 6512 . 2 ({𝑥𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)
53, 4syl 17 1 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1540   = wceq 1542  {cab 2710  {csn 4629  cio 6494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3954  df-in 3956  df-ss 3966  df-sn 4630  df-pr 4632  df-uni 4910  df-iota 6496
This theorem is referenced by:  iotauni  6519  iota1  6521  iotaexOLD  6524  iota4  6525  iota5  6527  iota5f  34693  iotain  43176  iotaexeu  43177  iotasbc  43178  iotaequ  43188  iotavalb  43189  pm14.24  43191  sbiota1  43193  aiotaval  45803
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