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Theorem aiotaval 47045
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 46976 . . . . 5 ∃!𝑧{𝑧} = {𝑦}
2 eqcom 2742 . . . . . 6 ({𝑦} = {𝑧} ↔ {𝑧} = {𝑦})
32eubii 2583 . . . . 5 (∃!𝑧{𝑦} = {𝑧} ↔ ∃!𝑧{𝑧} = {𝑦})
41, 3mpbir 231 . . . 4 ∃!𝑧{𝑦} = {𝑧}
5 eqeq1 2739 . . . . 5 ({𝑥𝜑} = {𝑦} → ({𝑥𝜑} = {𝑧} ↔ {𝑦} = {𝑧}))
65eubidv 2584 . . . 4 ({𝑥𝜑} = {𝑦} → (∃!𝑧{𝑥𝜑} = {𝑧} ↔ ∃!𝑧{𝑦} = {𝑧}))
74, 6mpbiri 258 . . 3 ({𝑥𝜑} = {𝑦} → ∃!𝑧{𝑥𝜑} = {𝑧})
8 absn 4650 . . 3 ({𝑥𝜑} = {𝑦} ↔ ∀𝑥(𝜑𝑥 = 𝑦))
9 reuabaiotaiota 47037 . . . 4 (∃!𝑧{𝑥𝜑} = {𝑧} ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
10 eqcom 2742 . . . 4 ((℩𝑥𝜑) = (℩'𝑥𝜑) ↔ (℩'𝑥𝜑) = (℩𝑥𝜑))
119, 10bitri 275 . . 3 (∃!𝑧{𝑥𝜑} = {𝑧} ↔ (℩'𝑥𝜑) = (℩𝑥𝜑))
127, 8, 113imtr3i 291 . 2 (∀𝑥(𝜑𝑥 = 𝑦) → (℩'𝑥𝜑) = (℩𝑥𝜑))
13 iotaval 6534 . 2 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
1412, 13eqtrd 2775 1 (∀𝑥(𝜑𝑥 = 𝑦) → (℩'𝑥𝜑) = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535   = wceq 1537  ∃!weu 2566  {cab 2712  {csn 4631  cio 6514  ℩'caiota 47033
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-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-sn 4632  df-pr 4634  df-uni 4913  df-int 4952  df-iota 6516  df-aiota 47035
This theorem is referenced by:  aiota0def  47046
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