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Theorem iota5 5108
Description: A method for computing iota. (Contributed by NM, 17-Sep-2013.)
Hypothesis
Ref Expression
iota5.1 ((𝜑𝐴𝑉) → (𝜓𝑥 = 𝐴))
Assertion
Ref Expression
iota5 ((𝜑𝐴𝑉) → (℩𝑥𝜓) = 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑉   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem iota5
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 iota5.1 . . 3 ((𝜑𝐴𝑉) → (𝜓𝑥 = 𝐴))
21alrimiv 1846 . 2 ((𝜑𝐴𝑉) → ∀𝑥(𝜓𝑥 = 𝐴))
3 eqeq2 2149 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
43bibi2d 231 . . . . . 6 (𝑦 = 𝐴 → ((𝜓𝑥 = 𝑦) ↔ (𝜓𝑥 = 𝐴)))
54albidv 1796 . . . . 5 (𝑦 = 𝐴 → (∀𝑥(𝜓𝑥 = 𝑦) ↔ ∀𝑥(𝜓𝑥 = 𝐴)))
6 eqeq2 2149 . . . . 5 (𝑦 = 𝐴 → ((℩𝑥𝜓) = 𝑦 ↔ (℩𝑥𝜓) = 𝐴))
75, 6imbi12d 233 . . . 4 (𝑦 = 𝐴 → ((∀𝑥(𝜓𝑥 = 𝑦) → (℩𝑥𝜓) = 𝑦) ↔ (∀𝑥(𝜓𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴)))
8 iotaval 5099 . . . 4 (∀𝑥(𝜓𝑥 = 𝑦) → (℩𝑥𝜓) = 𝑦)
97, 8vtoclg 2746 . . 3 (𝐴𝑉 → (∀𝑥(𝜓𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴))
109adantl 275 . 2 ((𝜑𝐴𝑉) → (∀𝑥(𝜓𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴))
112, 10mpd 13 1 ((𝜑𝐴𝑉) → (℩𝑥𝜓) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1329   = wceq 1331  wcel 1480  cio 5086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-sn 3533  df-pr 3534  df-uni 3737  df-iota 5088
This theorem is referenced by:  fsum3  11168
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