New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  tz6.12i GIF version

Theorem tz6.12i 5348
 Description: Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by set.mm contributors, 30-Apr-2004.) (Revised by set.mm contributors, 6-Apr-2007.)
Assertion
Ref Expression
tz6.12i (B → ((FA) = BAFB))

Proof of Theorem tz6.12i
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 tz6.12-2 5346 . . . . 5 ∃!y AFy → (FA) = )
21necon1ai 2558 . . . 4 ((FA) ≠ ∃!y AFy)
3 eqid 2353 . . . . 5 (FA) = (FA)
4 tz6.12c 5347 . . . . 5 (∃!y AFy → ((FA) = (FA) ↔ AF(FA)))
53, 4mpbii 202 . . . 4 (∃!y AFyAF(FA))
62, 5syl 15 . . 3 ((FA) ≠ AF(FA))
7 neeq1 2524 . . . 4 ((FA) = B → ((FA) ≠ B))
8 breq2 4643 . . . 4 ((FA) = B → (AF(FA) ↔ AFB))
97, 8imbi12d 311 . . 3 ((FA) = B → (((FA) ≠ AF(FA)) ↔ (BAFB)))
106, 9mpbii 202 . 2 ((FA) = B → (BAFB))
1110com12 27 1 (B → ((FA) = BAFB))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642  ∃!weu 2204   ≠ wne 2516  ∅c0 3550   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:  fvclss  5462
 Copyright terms: Public domain W3C validator