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Theorem csbie2g 3530
 Description: Conversion of implicit substitution to explicit class substitution. This version of csbie 3525 avoids a disjointness condition on 𝑥, 𝐴 and 𝑥, 𝐷 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 3500 . 2 𝐴 / 𝑥𝐵 = {𝑧[𝐴 / 𝑥]𝑧𝐵}
2 csbie2g.1 . . . . 5 (𝑥 = 𝑦𝐵 = 𝐶)
32eleq2d 2673 . . . 4 (𝑥 = 𝑦 → (𝑧𝐵𝑧𝐶))
4 csbie2g.2 . . . . 5 (𝑦 = 𝐴𝐶 = 𝐷)
54eleq2d 2673 . . . 4 (𝑦 = 𝐴 → (𝑧𝐶𝑧𝐷))
63, 5sbcie2g 3436 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝐵𝑧𝐷))
76abbi1dv 2730 . 2 (𝐴𝑉 → {𝑧[𝐴 / 𝑥]𝑧𝐵} = 𝐷)
81, 7syl5eq 2656 1 (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {cab 2596  [wsbc 3402  ⦋csb 3499 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175  df-sbc 3403  df-csb 3500 This theorem is referenced by: (None)
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