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Theorem eqer 5961
 Description: Equivalence relation involving equality of dependent classes A(x) and B(y). (Contributed by set.mm contributors, 17-Mar-2008.)
Hypotheses
Ref Expression
eqer.1 (x = yA = B)
eqer.2 R = {x, y A = B}
eqer.3 R V
Assertion
Ref Expression
eqer R Er V
Distinct variable groups:   y,A   x,B   x,y
Allowed substitution hints:   A(x)   B(y)   R(x,y)

Proof of Theorem eqer
Dummy variables v w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqer.3 . . . 4 R V
21a1i 10 . . 3 ( ⊤ → R V)
3 vvex 4109 . . . 4 V V
43a1i 10 . . 3 ( ⊤ → V V)
5 id 19 . . . . . 6 ([z / x]A = [w / x]A[z / x]A = [w / x]A)
65eqcomd 2358 . . . . 5 ([z / x]A = [w / x]A[w / x]A = [z / x]A)
7 eqer.1 . . . . . 6 (x = yA = B)
8 eqer.2 . . . . . 6 R = {x, y A = B}
97, 8eqerlem 5960 . . . . 5 (zRw[z / x]A = [w / x]A)
107, 8eqerlem 5960 . . . . 5 (wRz[w / x]A = [z / x]A)
116, 9, 103imtr4i 257 . . . 4 (zRwwRz)
12113ad2ant3 978 . . 3 (( ⊤ (z V w V) zRw) → wRz)
13 eqtr 2370 . . . . 5 (([z / x]A = [w / x]A [w / x]A = [v / x]A) → [z / x]A = [v / x]A)
147, 8eqerlem 5960 . . . . . 6 (wRv[w / x]A = [v / x]A)
159, 14anbi12i 678 . . . . 5 ((zRw wRv) ↔ ([z / x]A = [w / x]A [w / x]A = [v / x]A))
167, 8eqerlem 5960 . . . . 5 (zRv[z / x]A = [v / x]A)
1713, 15, 163imtr4i 257 . . . 4 ((zRw wRv) → zRv)
18173ad2ant3 978 . . 3 (( ⊤ (z V w V v V) (zRw wRv)) → zRv)
192, 4, 12, 18iserd 5942 . 2 ( ⊤ → R Er V)
2019trud 1323 1 R Er V
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934   ⊤ wtru 1316   = wceq 1642   ∈ wcel 1710  Vcvv 2859  [csb 3136  {copab 4622   class class class wbr 4639   Er cer 5898 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-csb 3137  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-opab 4623  df-br 4640  df-trans 5899  df-sym 5908  df-er 5909 This theorem is referenced by: (None)
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