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Theorem csbnestgf 3056
Description: Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
Assertion
Ref Expression
csbnestgf ((𝐴𝑉 ∧ ∀𝑦𝑥𝐶) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)

Proof of Theorem csbnestgf
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 elex 2700 . . 3 (𝐴𝑉𝐴 ∈ V)
2 df-csb 3007 . . . . . . 7 𝐵 / 𝑦𝐶 = {𝑧[𝐵 / 𝑦]𝑧𝐶}
32abeq2i 2251 . . . . . 6 (𝑧𝐵 / 𝑦𝐶[𝐵 / 𝑦]𝑧𝐶)
43sbcbii 2971 . . . . 5 ([𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶[𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐶)
5 nfcr 2274 . . . . . . 7 (𝑥𝐶 → Ⅎ𝑥 𝑧𝐶)
65alimi 1432 . . . . . 6 (∀𝑦𝑥𝐶 → ∀𝑦𝑥 𝑧𝐶)
7 sbcnestgf 3055 . . . . . 6 ((𝐴 ∈ V ∧ ∀𝑦𝑥 𝑧𝐶) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐶[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶))
86, 7sylan2 284 . . . . 5 ((𝐴 ∈ V ∧ ∀𝑦𝑥𝐶) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐶[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶))
94, 8syl5bb 191 . . . 4 ((𝐴 ∈ V ∧ ∀𝑦𝑥𝐶) → ([𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶))
109abbidv 2258 . . 3 ((𝐴 ∈ V ∧ ∀𝑦𝑥𝐶) → {𝑧[𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶} = {𝑧[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶})
111, 10sylan 281 . 2 ((𝐴𝑉 ∧ ∀𝑦𝑥𝐶) → {𝑧[𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶} = {𝑧[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶})
12 df-csb 3007 . 2 𝐴 / 𝑥𝐵 / 𝑦𝐶 = {𝑧[𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶}
13 df-csb 3007 . 2 𝐴 / 𝑥𝐵 / 𝑦𝐶 = {𝑧[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶}
1411, 12, 133eqtr4g 2198 1 ((𝐴𝑉 ∧ ∀𝑦𝑥𝐶) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1330   = wceq 1332  wnf 1437  wcel 1481  {cab 2126  wnfc 2269  Vcvv 2689  [wsbc 2912  csb 3006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sbc 2913  df-csb 3007
This theorem is referenced by:  csbnestg  3058  csbnest1g  3059
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