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Theorem tz6.12c 5116
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 1903 . . . 4 (∃!y A𝐹yy A𝐹y)
2 nfeu1 1884 . . . . . 6 y∃!y A𝐹y
3 nfv 1394 . . . . . 6 y A𝐹(𝐹A)
42, 3nfim 1437 . . . . 5 y(∃!y A𝐹yA𝐹(𝐹A))
5 tz6.12-1 5113 . . . . . . . 8 ((A𝐹y ∃!y A𝐹y) → (𝐹A) = y)
65expcom 109 . . . . . . 7 (∃!y A𝐹y → (A𝐹y → (𝐹A) = y))
7 breq2 3731 . . . . . . . 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 1458 . . . 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 1223  wex 1354  ∃!weu 1873   class class class wbr 3727  cfv 4817
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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-eu 1876  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-rex 2281  df-v 2528  df-sbc 2733  df-un 2890  df-sn 3345  df-pr 3346  df-op 3348  df-uni 3544  df-br 3728  df-iota 4782  df-fv 4825
This theorem is referenced by:  fnbrfvb  5127
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