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Theorem ecelqsg 5979
 Description: Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.)
Assertion
Ref Expression
ecelqsg ((R V B A) → [B]R (A / R))

Proof of Theorem ecelqsg
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eqid 2353 . . 3 [B]R = [B]R
2 eceq1 5962 . . . . 5 (x = B → [x]R = [B]R)
32eqeq2d 2364 . . . 4 (x = B → ([B]R = [x]R ↔ [B]R = [B]R))
43rspcev 2955 . . 3 ((B A [B]R = [B]R) → x A [B]R = [x]R)
51, 4mpan2 652 . 2 (B Ax A [B]R = [x]R)
6 ecexg 5949 . . . 4 (R V → [B]R V)
7 elqsg 5976 . . . 4 ([B]R V → ([B]R (A / R) ↔ x A [B]R = [x]R))
86, 7syl 15 . . 3 (R V → ([B]R (A / R) ↔ x A [B]R = [x]R))
98biimpar 471 . 2 ((R V x A [B]R = [x]R) → [B]R (A / R))
105, 9sylan2 460 1 ((R V B A) → [B]R (A / R))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  Vcvv 2859  [cec 5945   / cqs 5946 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-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-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  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-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-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-br 4640  df-ima 4727  df-ec 5947  df-qs 5951 This theorem is referenced by:  ecelqsi  5980
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