MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iotaval Structured version   Visualization version   GIF version

Theorem iotaval 6520
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 2129, ax-11 2146, ax-12 2166. (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 2793 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑥𝑥 = 𝑦})
2 df-sn 4631 . . 3 {𝑦} = {𝑥𝑥 = 𝑦}
31, 2eqtr4di 2783 . 2 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = {𝑦})
4 iotaval2 6517 . 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 1531   = wceq 1533  {cab 2702  {csn 4630  cio 6499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3463  df-un 3949  df-ss 3961  df-sn 4631  df-pr 4633  df-uni 4910  df-iota 6501
This theorem is referenced by:  iotauni  6524  iota1  6526  iotaexOLD  6529  iota4  6530  iota5  6532  iota5f  35449  iotain  43996  iotaexeu  43997  iotasbc  43998  iotaequ  44008  iotavalb  44009  pm14.24  44011  sbiota1  44013  aiotaval  46613
  Copyright terms: Public domain W3C validator