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Theorem iota2 5780
Description: The unique element such that 𝜑. (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
Hypothesis
Ref Expression
iota2.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
iota2 ((𝐴𝐵 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem iota2
StepHypRef Expression
1 elex 3185 . 2 (𝐴𝐵𝐴 ∈ V)
2 simpl 472 . . 3 ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → 𝐴 ∈ V)
3 simpr 476 . . 3 ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → ∃!𝑥𝜑)
4 iota2.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
54adantl 481 . . 3 (((𝐴 ∈ V ∧ ∃!𝑥𝜑) ∧ 𝑥 = 𝐴) → (𝜑𝜓))
6 nfv 1830 . . . 4 𝑥 𝐴 ∈ V
7 nfeu1 2468 . . . 4 𝑥∃!𝑥𝜑
86, 7nfan 1816 . . 3 𝑥(𝐴 ∈ V ∧ ∃!𝑥𝜑)
9 nfvd 1831 . . 3 ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → Ⅎ𝑥𝜓)
10 nfcvd 2752 . . 3 ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → 𝑥𝐴)
112, 3, 5, 8, 9, 10iota2df 5778 . 2 ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))
121, 11sylan 487 1 ((𝐴𝐵 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  ∃!weu 2458  Vcvv 3173  cio 5752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403  df-un 3545  df-sn 4126  df-pr 4128  df-uni 4368  df-iota 5754
This theorem is referenced by:  pczpre  15339  pcdiv  15344  rngurd  28913  bj-nuliota  32004  unirep  32471  ellimciota  38475
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