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Theorem aiotaval 43693
 Description: Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of (alternate) iota. (Contributed by AV, 24-Aug-2022.)
Assertion
Ref Expression
aiotaval (∀𝑥(𝜑𝑥 = 𝑦) → (℩'𝑥𝜑) = 𝑦)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem aiotaval
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eusnsn 43661 . . . . 5 ∃!𝑧{𝑧} = {𝑦}
2 eqcom 2805 . . . . . 6 ({𝑦} = {𝑧} ↔ {𝑧} = {𝑦})
32eubii 2645 . . . . 5 (∃!𝑧{𝑦} = {𝑧} ↔ ∃!𝑧{𝑧} = {𝑦})
41, 3mpbir 234 . . . 4 ∃!𝑧{𝑦} = {𝑧}
5 eqeq1 2802 . . . . 5 ({𝑥𝜑} = {𝑦} → ({𝑥𝜑} = {𝑧} ↔ {𝑦} = {𝑧}))
65eubidv 2647 . . . 4 ({𝑥𝜑} = {𝑦} → (∃!𝑧{𝑥𝜑} = {𝑧} ↔ ∃!𝑧{𝑦} = {𝑧}))
74, 6mpbiri 261 . . 3 ({𝑥𝜑} = {𝑦} → ∃!𝑧{𝑥𝜑} = {𝑧})
8 absn 4543 . . 3 ({𝑥𝜑} = {𝑦} ↔ ∀𝑥(𝜑𝑥 = 𝑦))
9 reuabaiotaiota 43687 . . . 4 (∃!𝑧{𝑥𝜑} = {𝑧} ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
10 eqcom 2805 . . . 4 ((℩𝑥𝜑) = (℩'𝑥𝜑) ↔ (℩'𝑥𝜑) = (℩𝑥𝜑))
119, 10bitri 278 . . 3 (∃!𝑧{𝑥𝜑} = {𝑧} ↔ (℩'𝑥𝜑) = (℩𝑥𝜑))
127, 8, 113imtr3i 294 . 2 (∀𝑥(𝜑𝑥 = 𝑦) → (℩'𝑥𝜑) = (℩𝑥𝜑))
13 iotaval 6299 . 2 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
1412, 13eqtrd 2833 1 (∀𝑥(𝜑𝑥 = 𝑦) → (℩'𝑥𝜑) = 𝑦)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   = wceq 1538  ∃!weu 2628  {cab 2776  {csn 4525  ℩cio 6282  ℩'caiota 43683 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-uni 4802  df-int 4840  df-iota 6284  df-aiota 43685 This theorem is referenced by:  aiota0def  43694
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