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Theorem csbco3g 2904
 Description: Composition of two class substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
Hypothesis
Ref Expression
sbcco3g.1 (𝑥 = 𝐴𝐵 = 𝐶)
Assertion
Ref Expression
csbco3g (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝐷 = 𝐶 / 𝑦𝐷)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑦)   𝐷(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem csbco3g
StepHypRef Expression
1 csbnestg 2900 . 2 (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝐷 = 𝐴 / 𝑥𝐵 / 𝑦𝐷)
2 elex 2566 . . . 4 (𝐴𝑉𝐴 ∈ V)
3 nfcvd 2179 . . . . 5 (𝐴 ∈ V → 𝑥𝐶)
4 sbcco3g.1 . . . . 5 (𝑥 = 𝐴𝐵 = 𝐶)
53, 4csbiegf 2890 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐵 = 𝐶)
62, 5syl 14 . . 3 (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐶)
76csbeq1d 2858 . 2 (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝐷 = 𝐶 / 𝑦𝐷)
81, 7eqtrd 2072 1 (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝐷 = 𝐶 / 𝑦𝐷)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393  Vcvv 2557  ⦋csb 2852 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-csb 2853 This theorem is referenced by: (None)
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