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Theorem iotan0 6318
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 3088 . . . . . 6 ((𝐴 = (℩𝑥𝜑) ∧ 𝐴 ≠ ∅) → (℩𝑥𝜑) ≠ ∅)
21expcom 417 . . . . 5 (𝐴 ≠ ∅ → (𝐴 = (℩𝑥𝜑) → (℩𝑥𝜑) ≠ ∅))
3 iotanul 6306 . . . . . 6 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
43necon1ai 3034 . . . . 5 ((℩𝑥𝜑) ≠ ∅ → ∃!𝑥𝜑)
52, 4syl6 35 . . . 4 (𝐴 ≠ ∅ → (𝐴 = (℩𝑥𝜑) → ∃!𝑥𝜑))
65a1i 11 . . 3 (𝐴𝑉 → (𝐴 ≠ ∅ → (𝐴 = (℩𝑥𝜑) → ∃!𝑥𝜑)))
763imp 1108 . 2 ((𝐴𝑉𝐴 ≠ ∅ ∧ 𝐴 = (℩𝑥𝜑)) → ∃!𝑥𝜑)
8 eqcom 2828 . . . . 5 (𝐴 = (℩𝑥𝜑) ↔ (℩𝑥𝜑) = 𝐴)
9 iotan0.1 . . . . . . 7 (𝑥 = 𝐴 → (𝜑𝜓))
109iota2 6317 . . . . . 6 ((𝐴𝑉 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))
1110biimprd 251 . . . . 5 ((𝐴𝑉 ∧ ∃!𝑥𝜑) → ((℩𝑥𝜑) = 𝐴𝜓))
128, 11syl5bi 245 . . . 4 ((𝐴𝑉 ∧ ∃!𝑥𝜑) → (𝐴 = (℩𝑥𝜑) → 𝜓))
1312impancom 455 . . 3 ((𝐴𝑉𝐴 = (℩𝑥𝜑)) → (∃!𝑥𝜑𝜓))
14133adant2 1128 . 2 ((𝐴𝑉𝐴 ≠ ∅ ∧ 𝐴 = (℩𝑥𝜑)) → (∃!𝑥𝜑𝜓))
157, 14mpd 15 1 ((𝐴𝑉𝐴 ≠ ∅ ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  ∃!weu 2653  wne 3007  c0 4266  cio 6285
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793
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 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-sn 4541  df-pr 4543  df-uni 4812  df-iota 6287
This theorem is referenced by:  sgrpidmnd  17895
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