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Theorem csbima12g 4793
Description: Move class substitution in and out of the image of a function. (Contributed by FL, 15-Dec-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
Assertion
Ref Expression
csbima12g (𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))

Proof of Theorem csbima12g
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 2936 . . 3 (𝑦 = 𝐴𝑦 / 𝑥(𝐹𝐵) = 𝐴 / 𝑥(𝐹𝐵))
2 csbeq1 2936 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)
3 csbeq1 2936 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
42, 3imaeq12d 4775 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
51, 4eqeq12d 2102 . 2 (𝑦 = 𝐴 → (𝑦 / 𝑥(𝐹𝐵) = (𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵) ↔ 𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵)))
6 vex 2622 . . 3 𝑦 ∈ V
7 nfcsb1v 2963 . . . 4 𝑥𝑦 / 𝑥𝐹
8 nfcsb1v 2963 . . . 4 𝑥𝑦 / 𝑥𝐵
97, 8nfima 4782 . . 3 𝑥(𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵)
10 csbeq1a 2941 . . . 4 (𝑥 = 𝑦𝐹 = 𝑦 / 𝑥𝐹)
11 csbeq1a 2941 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
1210, 11imaeq12d 4775 . . 3 (𝑥 = 𝑦 → (𝐹𝐵) = (𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵))
136, 9, 12csbief 2972 . 2 𝑦 / 𝑥(𝐹𝐵) = (𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵)
145, 13vtoclg 2679 1 (𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  wcel 1438  csb 2933  cima 4441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-cnv 4446  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451
This theorem is referenced by: (None)
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