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Theorem reiota2 4368
 Description: A condition allowing us to represent "the unique element in A such that φ " with a class expression B. (Contributed by Scott Fenton, 7-Jan-2018.)
Hypothesis
Ref Expression
reiota2.1 (x = B → (φψ))
Assertion
Ref Expression
reiota2 ((B A ∃!x A φ) → (ψ ↔ (℩x(x A φ)) = B))
Distinct variable groups:   x,A   x,B   ψ,x
Allowed substitution hint:   φ(x)

Proof of Theorem reiota2
StepHypRef Expression
1 simpl 443 . . 3 ((B A ∃!x A φ) → B A)
21biantrurd 494 . 2 ((B A ∃!x A φ) → (ψ ↔ (B A ψ)))
3 df-reu 2621 . . 3 (∃!x A φ∃!x(x A φ))
4 eleq1 2413 . . . . 5 (x = B → (x AB A))
5 reiota2.1 . . . . 5 (x = B → (φψ))
64, 5anbi12d 691 . . . 4 (x = B → ((x A φ) ↔ (B A ψ)))
76iota2 4367 . . 3 ((B A ∃!x(x A φ)) → ((B A ψ) ↔ (℩x(x A φ)) = B))
83, 7sylan2b 461 . 2 ((B A ∃!x A φ) → ((B A ψ) ↔ (℩x(x A φ)) = B))
92, 8bitrd 244 1 ((B A ∃!x A φ) → (ψ ↔ (℩x(x A φ)) = B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  ∃!wreu 2616  ℩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-reu 2621  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:  ncfinprop  4474  tfinprop  4489  eqtc  6161
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