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Theorem iota2 4367
 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 2867 . 2 (A BA V)
2 simpl 443 . . 3 ((A V ∃!xφ) → A V)
3 simpr 447 . . 3 ((A V ∃!xφ) → ∃!xφ)
4 iota2.1 . . . 4 (x = A → (φψ))
54adantl 452 . . 3 (((A V ∃!xφ) x = A) → (φψ))
6 nfv 1619 . . . 4 x A V
7 nfeu1 2214 . . . 4 x∃!xφ
86, 7nfan 1824 . . 3 x(A V ∃!xφ)
9 nfvd 1620 . . 3 ((A V ∃!xφ) → Ⅎxψ)
10 nfcvd 2490 . . 3 ((A V ∃!xφ) → xA)
112, 3, 5, 8, 9, 10iota2df 4365 . 2 ((A V ∃!xφ) → (ψ ↔ (℩xφ) = A))
121, 11sylan 457 1 ((A B ∃!xφ) → (ψ ↔ (℩xφ) = A))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  Vcvv 2859  ℩cio 4337 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-uni 3892  df-iota 4339 This theorem is referenced by:  reiota2  4368  tz6.12-1  5344
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