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Theorem tz6.12c 5347
 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) = BAFB))
Distinct variable groups:   y,F   y,A
Allowed substitution hint:   B(y)

Proof of Theorem tz6.12c
StepHypRef Expression
1 euex 2227 . . . 4 (∃!y AFyy AFy)
2 nfeu1 2214 . . . . . 6 y∃!y AFy
3 nfv 1619 . . . . . 6 y AF(FA)
42, 3nfim 1813 . . . . 5 y(∃!y AFyAF(FA))
5 tz6.12-1 5344 . . . . . . . 8 ((AFy ∃!y AFy) → (FA) = y)
65expcom 424 . . . . . . 7 (∃!y AFy → (AFy → (FA) = y))
7 breq2 4643 . . . . . . . 8 ((FA) = y → (AF(FA) ↔ AFy))
87biimprd 214 . . . . . . 7 ((FA) = y → (AFyAF(FA)))
96, 8syli 33 . . . . . 6 (∃!y AFy → (AFyAF(FA)))
109com12 27 . . . . 5 (AFy → (∃!y AFyAF(FA)))
114, 10exlimi 1803 . . . 4 (y AFy → (∃!y AFyAF(FA)))
121, 11mpcom 32 . . 3 (∃!y AFyAF(FA))
13 breq2 4643 . . 3 ((FA) = B → (AF(FA) ↔ AFB))
1412, 13syl5ibcom 211 . 2 (∃!y AFy → ((FA) = BAFB))
15 tz6.12-1 5344 . . 3 ((AFB ∃!y AFy) → (FA) = B)
1615expcom 424 . 2 (∃!y AFy → (AFB → (FA) = B))
1714, 16impbid 183 1 (∃!y AFy → ((FA) = BAFB))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∃wex 1541   = wceq 1642  ∃!weu 2204   class class class wbr 4639   ‘cfv 4781 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-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-br 4640  df-fv 4795 This theorem is referenced by:  tz6.12i  5348  fnbrfvb  5358
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