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Theorem sbcnestg 4374
Description: Nest the composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
Assertion
Ref Expression
sbcnestg (𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem sbcnestg
StepHypRef Expression
1 nfv 1906 . . 3 𝑥𝜑
21ax-gen 1787 . 2 𝑦𝑥𝜑
3 sbcnestgf 4372 . 2 ((𝐴𝑉 ∧ ∀𝑦𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
42, 3mpan2 687 1 (𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1526  wnf 1775  wcel 2105  [wsbc 3769  csb 3880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-13 2381  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-sbc 3770  df-csb 3881
This theorem is referenced by:  sbcco3g  4376
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