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Theorem tz6.12c 5146
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 1927 . . . 4 (∃!y A𝐹yy A𝐹y)
2 nfeu1 1908 . . . . . 6 y∃!y A𝐹y
3 nfv 1418 . . . . . 6 y A𝐹(𝐹A)
42, 3nfim 1461 . . . . 5 y(∃!y A𝐹yA𝐹(𝐹A))
5 tz6.12-1 5143 . . . . . . . 8 ((A𝐹y ∃!y A𝐹y) → (𝐹A) = y)
65expcom 109 . . . . . . 7 (∃!y A𝐹y → (A𝐹y → (𝐹A) = y))
7 breq2 3759 . . . . . . . 8 ((𝐹A) = y → (A𝐹(𝐹A) ↔ A𝐹y))
87biimprd 147 . . . . . . 7 ((𝐹A) = y → (A𝐹yA𝐹(𝐹A)))
96, 8syli 33 . . . . . 6 (∃!y A𝐹y → (A𝐹yA𝐹(𝐹A)))
109com12 27 . . . . 5 (A𝐹y → (∃!y A𝐹yA𝐹(𝐹A)))
114, 10exlimi 1482 . . . 4 (y A𝐹y → (∃!y A𝐹yA𝐹(𝐹A)))
121, 11mpcom 32 . . 3 (∃!y A𝐹yA𝐹(𝐹A))
1312, 7syl5ibcom 144 . 2 (∃!y A𝐹y → ((𝐹A) = yA𝐹y))
1413, 6impbid 120 1 (∃!y A𝐹y → ((𝐹A) = yA𝐹y))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242  wex 1378  ∃!weu 1897   class class class wbr 3755  cfv 4845
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-op 3376  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853
This theorem is referenced by:  fnbrfvb  5157
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