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Theorem iotaval 6534
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 2139, ax-11 2155, ax-12 2175. (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 2805 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑥𝑥 = 𝑦})
2 df-sn 4632 . . 3 {𝑦} = {𝑥𝑥 = 𝑦}
31, 2eqtr4di 2793 . 2 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑦})
4 iotaval2 6531 . 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 206  wal 1535   = wceq 1537  {cab 2712  {csn 4631  cio 6514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-ss 3980  df-sn 4632  df-pr 4634  df-uni 4913  df-iota 6516
This theorem is referenced by:  iotauni  6538  iota1  6540  iotaexOLD  6543  iota4  6544  iota5  6546  iota5f  35704  iotain  44413  iotaexeu  44414  iotasbc  44415  iotaequ  44425  iotavalb  44426  pm14.24  44428  sbiota1  44430  aiotaval  47045
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