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Theorem csbnestgf 3973
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 3201 . . 3 (𝐴𝑉𝐴 ∈ V)
2 df-csb 3519 . . . . . . 7 𝐵 / 𝑦𝐶 = {𝑧[𝐵 / 𝑦]𝑧𝐶}
32abeq2i 2732 . . . . . 6 (𝑧𝐵 / 𝑦𝐶[𝐵 / 𝑦]𝑧𝐶)
43sbcbii 3477 . . . . 5 ([𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶[𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐶)
5 nfcr 2753 . . . . . . 7 (𝑥𝐶 → Ⅎ𝑥 𝑧𝐶)
65alimi 1736 . . . . . 6 (∀𝑦𝑥𝐶 → ∀𝑦𝑥 𝑧𝐶)
7 sbcnestgf 3972 . . . . . 6 ((𝐴 ∈ V ∧ ∀𝑦𝑥 𝑧𝐶) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐶[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶))
86, 7sylan2 491 . . . . 5 ((𝐴 ∈ V ∧ ∀𝑦𝑥𝐶) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧𝐶[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶))
94, 8syl5bb 272 . . . 4 ((𝐴 ∈ V ∧ ∀𝑦𝑥𝐶) → ([𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶))
109abbidv 2738 . . 3 ((𝐴 ∈ V ∧ ∀𝑦𝑥𝐶) → {𝑧[𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶} = {𝑧[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶})
111, 10sylan 488 . 2 ((𝐴𝑉 ∧ ∀𝑦𝑥𝐶) → {𝑧[𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶} = {𝑧[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶})
12 df-csb 3519 . 2 𝐴 / 𝑥𝐵 / 𝑦𝐶 = {𝑧[𝐴 / 𝑥]𝑧𝐵 / 𝑦𝐶}
13 df-csb 3519 . 2 𝐴 / 𝑥𝐵 / 𝑦𝐶 = {𝑧[𝐴 / 𝑥𝐵 / 𝑦]𝑧𝐶}
1411, 12, 133eqtr4g 2680 1 ((𝐴𝑉 ∧ ∀𝑦𝑥𝐶) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wnf 1705  wcel 1987  {cab 2607  wnfc 2748  Vcvv 3189  [wsbc 3421  csb 3518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3191  df-sbc 3422  df-csb 3519
This theorem is referenced by:  csbnestg  3975  csbnest1g  3978
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