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Theorem iotan0 6550
Description: Representation of "the unique element such that 𝜑 " with a class expression 𝐴 which is not the empty set (that means that "the unique element such that 𝜑 " exists). (Contributed by AV, 30-Jan-2024.)
Hypothesis
Ref Expression
iotan0.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
iotan0 ((𝐴𝑉𝐴 ≠ ∅ ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓)
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem iotan0
StepHypRef Expression
1 pm13.18 3021 . . . . . 6 ((𝐴 = (℩𝑥𝜑) ∧ 𝐴 ≠ ∅) → (℩𝑥𝜑) ≠ ∅)
21expcom 413 . . . . 5 (𝐴 ≠ ∅ → (𝐴 = (℩𝑥𝜑) → (℩𝑥𝜑) ≠ ∅))
3 iotanul 6538 . . . . . 6 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
43necon1ai 2967 . . . . 5 ((℩𝑥𝜑) ≠ ∅ → ∃!𝑥𝜑)
52, 4syl6 35 . . . 4 (𝐴 ≠ ∅ → (𝐴 = (℩𝑥𝜑) → ∃!𝑥𝜑))
65a1i 11 . . 3 (𝐴𝑉 → (𝐴 ≠ ∅ → (𝐴 = (℩𝑥𝜑) → ∃!𝑥𝜑)))
763imp 1110 . 2 ((𝐴𝑉𝐴 ≠ ∅ ∧ 𝐴 = (℩𝑥𝜑)) → ∃!𝑥𝜑)
8 eqcom 2743 . . . . 5 (𝐴 = (℩𝑥𝜑) ↔ (℩𝑥𝜑) = 𝐴)
9 iotan0.1 . . . . . . 7 (𝑥 = 𝐴 → (𝜑𝜓))
109iota2 6549 . . . . . 6 ((𝐴𝑉 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))
1110biimprd 248 . . . . 5 ((𝐴𝑉 ∧ ∃!𝑥𝜑) → ((℩𝑥𝜑) = 𝐴𝜓))
128, 11biimtrid 242 . . . 4 ((𝐴𝑉 ∧ ∃!𝑥𝜑) → (𝐴 = (℩𝑥𝜑) → 𝜓))
1312impancom 451 . . 3 ((𝐴𝑉𝐴 = (℩𝑥𝜑)) → (∃!𝑥𝜑𝜓))
14133adant2 1131 . 2 ((𝐴𝑉𝐴 ≠ ∅ ∧ 𝐴 = (℩𝑥𝜑)) → (∃!𝑥𝜑𝜓))
157, 14mpd 15 1 ((𝐴𝑉𝐴 ≠ ∅ ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  ∃!weu 2567  wne 2939  c0 4332  cio 6511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-sn 4626  df-pr 4628  df-uni 4907  df-iota 6513
This theorem is referenced by:  sgrpidmnd  18753
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