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Theorem iotaval 6464
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 2137, ax-11 2154, ax-12 2171. (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 abbi1 2804 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑥𝑥 = 𝑦})
2 df-sn 4585 . . 3 {𝑦} = {𝑥𝑥 = 𝑦}
31, 2eqtr4di 2794 . 2 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑦})
4 iotaval2 6461 . 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 1539   = wceq 1541  {cab 2713  {csn 4584  cio 6443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3445  df-un 3913  df-in 3915  df-ss 3925  df-sn 4585  df-pr 4587  df-uni 4864  df-iota 6445
This theorem is referenced by:  iotauni  6468  iota1  6470  iotaexOLD  6473  iota4  6474  iota5  6476  iota5f  34164  iotain  42639  iotaexeu  42640  iotasbc  42641  iotaequ  42651  iotavalb  42652  pm14.24  42654  sbiota1  42656  aiotaval  45259
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