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Theorem csbie2g 3854
 Description: Conversion of implicit substitution to explicit class substitution. This version of csbie 3849 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 3818 . 2 𝐴 / 𝑥𝐵 = {𝑧[𝐴 / 𝑥]𝑧𝐵}
2 csbie2g.1 . . . . 5 (𝑥 = 𝑦𝐵 = 𝐶)
32eleq2d 2870 . . . 4 (𝑥 = 𝑦 → (𝑧𝐵𝑧𝐶))
4 csbie2g.2 . . . . 5 (𝑦 = 𝐴𝐶 = 𝐷)
54eleq2d 2870 . . . 4 (𝑦 = 𝐴 → (𝑧𝐶𝑧𝐷))
63, 5sbcie2g 3745 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝐵𝑧𝐷))
76abbi1dv 2923 . 2 (𝐴𝑉 → {𝑧[𝐴 / 𝑥]𝑧𝐵} = 𝐷)
81, 7syl5eq 2845 1 (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1525   ∈ wcel 2083  {cab 2777  [wsbc 3711  ⦋csb 3817 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-12 2143  ax-ext 2771 This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-sbc 3712  df-csb 3818 This theorem is referenced by: (None)
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