Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  iota2 GIF version

Theorem iota2 5114
 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 2697 . 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 275 . . 3 (((𝐴 ∈ V ∧ ∃!𝑥𝜑) ∧ 𝑥 = 𝐴) → (𝜑𝜓))
6 nfv 1508 . . . 4 𝑥 𝐴 ∈ V
7 nfeu1 2010 . . . 4 𝑥∃!𝑥𝜑
86, 7nfan 1544 . . 3 𝑥(𝐴 ∈ V ∧ ∃!𝑥𝜑)
9 nfvd 1509 . . 3 ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → Ⅎ𝑥𝜓)
10 nfcvd 2282 . . 3 ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → 𝑥𝐴)
112, 3, 5, 8, 9, 10iota2df 5112 . 2 ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))
121, 11sylan 281 1 ((𝐴𝐵 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  ∃!weu 1999  Vcvv 2686  ℩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-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  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: (None)
 Copyright terms: Public domain W3C validator