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Theorem iota2 4921
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 2583 . 2 (𝐴𝐵𝐴 ∈ V)
2 simpl 106 . . 3 ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → 𝐴 ∈ V)
3 simpr 107 . . 3 ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → ∃!𝑥𝜑)
4 iota2.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
54adantl 266 . . 3 (((𝐴 ∈ V ∧ ∃!𝑥𝜑) ∧ 𝑥 = 𝐴) → (𝜑𝜓))
6 nfv 1437 . . . 4 𝑥 𝐴 ∈ V
7 nfeu1 1927 . . . 4 𝑥∃!𝑥𝜑
86, 7nfan 1473 . . 3 𝑥(𝐴 ∈ V ∧ ∃!𝑥𝜑)
9 nfvd 1438 . . 3 ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → Ⅎ𝑥𝜓)
10 nfcvd 2195 . . 3 ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → 𝑥𝐴)
112, 3, 5, 8, 9, 10iota2df 4919 . 2 ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))
121, 11sylan 271 1 ((𝐴𝐵 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  ∃!weu 1916  Vcvv 2574  cio 4893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-sn 3409  df-pr 3410  df-uni 3609  df-iota 4895
This theorem is referenced by: (None)
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