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Theorem csbrng 4966
 Description: Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.)
Assertion
Ref Expression
csbrng (A V[A / x]ran B = ran [A / x]B)

Proof of Theorem csbrng
Dummy variables w y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 csbabg 3197 . . 3 (A V[A / x]{y ww, y B} = {y A / xww, y B})
2 sbcexg 3096 . . . . 5 (A V → ([̣A / xww, y BwA / xw, y B))
3 sbcel2g 3157 . . . . . 6 (A V → ([̣A / xw, y Bw, y [A / x]B))
43exbidv 1626 . . . . 5 (A V → (wA / xw, y Bww, y [A / x]B))
52, 4bitrd 244 . . . 4 (A V → ([̣A / xww, y Bww, y [A / x]B))
65abbidv 2467 . . 3 (A V → {y A / xww, y B} = {y ww, y [A / x]B})
71, 6eqtrd 2385 . 2 (A V[A / x]{y ww, y B} = {y ww, y [A / x]B})
8 dfrn3 4903 . . 3 ran B = {y ww, y B}
98csbeq2i 3162 . 2 [A / x]ran B = [A / x]{y ww, y B}
10 dfrn3 4903 . 2 ran [A / x]B = {y ww, y [A / x]B}
117, 9, 103eqtr4g 2410 1 (A V[A / x]ran B = ran [A / x]B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  [̣wsbc 3046  [csb 3136  ⟨cop 4561  ran crn 4773 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-br 4640  df-ima 4727  df-rn 4786 This theorem is referenced by: (None)
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