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Theorem iotaval 6532
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 2141, ax-11 2157, ax-12 2177. (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 2807 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑥𝑥 = 𝑦})
2 df-sn 4627 . . 3 {𝑦} = {𝑥𝑥 = 𝑦}
31, 2eqtr4di 2795 . 2 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑦})
4 iotaval2 6529 . 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 1538   = wceq 1540  {cab 2714  {csn 4626  cio 6512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-ss 3968  df-sn 4627  df-pr 4629  df-uni 4908  df-iota 6514
This theorem is referenced by:  iotauni  6536  iota1  6538  iotaexOLD  6541  iota4  6542  iota5  6544  iota5f  35724  iotain  44436  iotaexeu  44437  iotasbc  44438  iotaequ  44448  iotavalb  44449  pm14.24  44451  sbiota1  44453  aiotaval  47107
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