Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  tz6.12c GIF version

Theorem tz6.12c 5369
 Description: Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
Assertion
Ref Expression
tz6.12c (∃!𝑦 𝐴𝐹𝑦 → ((𝐹𝐴) = 𝑦𝐴𝐹𝑦))
Distinct variable groups:   𝑦,𝐹   𝑦,𝐴

Proof of Theorem tz6.12c
StepHypRef Expression
1 euex 1985 . . . 4 (∃!𝑦 𝐴𝐹𝑦 → ∃𝑦 𝐴𝐹𝑦)
2 nfeu1 1966 . . . . . 6 𝑦∃!𝑦 𝐴𝐹𝑦
3 nfv 1473 . . . . . 6 𝑦 𝐴𝐹(𝐹𝐴)
42, 3nfim 1516 . . . . 5 𝑦(∃!𝑦 𝐴𝐹𝑦𝐴𝐹(𝐹𝐴))
5 tz6.12-1 5366 . . . . . . . 8 ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦)
65expcom 115 . . . . . . 7 (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 → (𝐹𝐴) = 𝑦))
7 breq2 3871 . . . . . . . 8 ((𝐹𝐴) = 𝑦 → (𝐴𝐹(𝐹𝐴) ↔ 𝐴𝐹𝑦))
87biimprd 157 . . . . . . 7 ((𝐹𝐴) = 𝑦 → (𝐴𝐹𝑦𝐴𝐹(𝐹𝐴)))
96, 8syli 37 . . . . . 6 (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦𝐴𝐹(𝐹𝐴)))
109com12 30 . . . . 5 (𝐴𝐹𝑦 → (∃!𝑦 𝐴𝐹𝑦𝐴𝐹(𝐹𝐴)))
114, 10exlimi 1537 . . . 4 (∃𝑦 𝐴𝐹𝑦 → (∃!𝑦 𝐴𝐹𝑦𝐴𝐹(𝐹𝐴)))
121, 11mpcom 36 . . 3 (∃!𝑦 𝐴𝐹𝑦𝐴𝐹(𝐹𝐴))
1312, 7syl5ibcom 154 . 2 (∃!𝑦 𝐴𝐹𝑦 → ((𝐹𝐴) = 𝑦𝐴𝐹𝑦))
1413, 6impbid 128 1 (∃!𝑦 𝐴𝐹𝑦 → ((𝐹𝐴) = 𝑦𝐴𝐹𝑦))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1296  ∃wex 1433  ∃!weu 1955   class class class wbr 3867  ‘cfv 5049 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077 This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rex 2376  df-v 2635  df-sbc 2855  df-un 3017  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-iota 5014  df-fv 5057 This theorem is referenced by:  fnbrfvb  5380
 Copyright terms: Public domain W3C validator