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Theorem csbnestgf 2955
 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 2611 . . 3 (𝐴𝑉𝐴 ∈ V)
2 df-csb 2910 . . . . . . 7 𝐵 / 𝑦𝐶 = {𝑧[𝐵 / 𝑦]𝑧𝐶}
32abeq2i 2190 . . . . . 6 (𝑧𝐵 / 𝑦𝐶[𝐵 / 𝑦]𝑧𝐶)
43sbcbii 2874 . . . . 5 ([𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶[𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐶)
5 nfcr 2212 . . . . . . 7 (𝑥𝐶 → Ⅎ𝑥 𝑧𝐶)
65alimi 1385 . . . . . 6 (∀𝑦𝑥𝐶 → ∀𝑦𝑥 𝑧𝐶)
7 sbcnestgf 2954 . . . . . 6 ((𝐴 ∈ V ∧ ∀𝑦𝑥 𝑧𝐶) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐶[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶))
86, 7sylan2 280 . . . . 5 ((𝐴 ∈ V ∧ ∀𝑦𝑥𝐶) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐶[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶))
94, 8syl5bb 190 . . . 4 ((𝐴 ∈ V ∧ ∀𝑦𝑥𝐶) → ([𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶))
109abbidv 2197 . . 3 ((𝐴 ∈ V ∧ ∀𝑦𝑥𝐶) → {𝑧[𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶} = {𝑧[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶})
111, 10sylan 277 . 2 ((𝐴𝑉 ∧ ∀𝑦𝑥𝐶) → {𝑧[𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶} = {𝑧[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶})
12 df-csb 2910 . 2 𝐴 / 𝑥𝐵 / 𝑦𝐶 = {𝑧[𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶}
13 df-csb 2910 . 2 𝐴 / 𝑥𝐵 / 𝑦𝐶 = {𝑧[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶}
1411, 12, 133eqtr4g 2139 1 ((𝐴𝑉 ∧ ∀𝑦𝑥𝐶) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103  ∀wal 1283   = wceq 1285  Ⅎwnf 1390   ∈ wcel 1434  {cab 2068  Ⅎwnfc 2207  Vcvv 2602  [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-3an 922  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:  csbnestg  2957  csbnest1g  2958
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