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Theorem csbie2g 2953
Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 2849 avoids a disjointness condition on 𝑥 and 𝐴 by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.)
Hypotheses
Ref Expression
csbie2g.1 (𝑥 = 𝑦𝐵 = 𝐶)
csbie2g.2 (𝑦 = 𝐴𝐶 = 𝐷)
Assertion
Ref Expression
csbie2g (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐷)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑦,𝐵   𝑥,𝐶   𝑦,𝐷
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑦)   𝐷(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem csbie2g
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-csb 2910 . 2 𝐴 / 𝑥𝐵 = {𝑧[𝐴 / 𝑥]𝑧𝐵}
2 csbie2g.1 . . . . 5 (𝑥 = 𝑦𝐵 = 𝐶)
32eleq2d 2149 . . . 4 (𝑥 = 𝑦 → (𝑧𝐵𝑧𝐶))
4 csbie2g.2 . . . . 5 (𝑦 = 𝐴𝐶 = 𝐷)
54eleq2d 2149 . . . 4 (𝑦 = 𝐴 → (𝑧𝐶𝑧𝐷))
63, 5sbcie2g 2848 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝐵𝑧𝐷))
76abbi1dv 2199 . 2 (𝐴𝑉 → {𝑧[𝐴 / 𝑥]𝑧𝐵} = 𝐷)
81, 7syl5eq 2126 1 (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wcel 1434  {cab 2068  [wsbc 2816  csb 2909
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-sbc 2817  df-csb 2910
This theorem is referenced by: (None)
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