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Theorem iota2 4836
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 ((A B ∃!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 2560 . 2 (A BA V)
2 simpl 102 . . 3 ((A V ∃!xφ) → A V)
3 simpr 103 . . 3 ((A V ∃!xφ) → ∃!xφ)
4 iota2.1 . . . 4 (x = A → (φψ))
54adantl 262 . . 3 (((A V ∃!xφ) x = A) → (φψ))
6 nfv 1418 . . . 4 x A V
7 nfeu1 1908 . . . 4 x∃!xφ
86, 7nfan 1454 . . 3 x(A V ∃!xφ)
9 nfvd 1419 . . 3 ((A V ∃!xφ) → Ⅎxψ)
10 nfcvd 2176 . . 3 ((A V ∃!xφ) → xA)
112, 3, 5, 8, 9, 10iota2df 4834 . 2 ((A V ∃!xφ) → (ψ ↔ (℩xφ) = A))
121, 11sylan 267 1 ((A B ∃!xφ) → (ψ ↔ (℩xφ) = A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  ∃!weu 1897  Vcvv 2551  cio 4808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-sn 3373  df-pr 3374  df-uni 3572  df-iota 4810
This theorem is referenced by: (None)
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