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Theorem iota2 5040
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 2644 . 2 (𝐴𝐵𝐴 ∈ V)
2 simpl 108 . . 3 ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → 𝐴 ∈ V)
3 simpr 109 . . 3 ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → ∃!𝑥𝜑)
4 iota2.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
54adantl 272 . . 3 (((𝐴 ∈ V ∧ ∃!𝑥𝜑) ∧ 𝑥 = 𝐴) → (𝜑𝜓))
6 nfv 1473 . . . 4 𝑥 𝐴 ∈ V
7 nfeu1 1966 . . . 4 𝑥∃!𝑥𝜑
86, 7nfan 1509 . . 3 𝑥(𝐴 ∈ V ∧ ∃!𝑥𝜑)
9 nfvd 1474 . . 3 ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → Ⅎ𝑥𝜓)
10 nfcvd 2236 . . 3 ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → 𝑥𝐴)
112, 3, 5, 8, 9, 10iota2df 5038 . 2 ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))
121, 11sylan 278 1 ((𝐴𝐵 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1296  wcel 1445  ∃!weu 1955  Vcvv 2633  cio 5012
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rex 2376  df-v 2635  df-sbc 2855  df-un 3017  df-sn 3472  df-pr 3473  df-uni 3676  df-iota 5014
This theorem is referenced by: (None)
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