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Theorem csbnest1g 4431
Description: Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
Assertion
Ref Expression
csbnest1g (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑥𝐶)

Proof of Theorem csbnest1g
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfcsb1v 3914 . . . 4 𝑥𝑦 / 𝑥𝐶
21ax-gen 1789 . . 3 𝑦𝑥𝑦 / 𝑥𝐶
3 csbnestgfw 4421 . . 3 ((𝐴𝑉 ∧ ∀𝑦𝑥𝑦 / 𝑥𝐶) → 𝐴 / 𝑥𝐵 / 𝑦𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝑦 / 𝑥𝐶)
42, 3mpan2 689 . 2 (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝑦 / 𝑥𝐶)
5 csbcow 3904 . . 3 𝐵 / 𝑦𝑦 / 𝑥𝐶 = 𝐵 / 𝑥𝐶
65csbeq2i 3897 . 2 𝐴 / 𝑥𝐵 / 𝑦𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑥𝐶
7 csbcow 3904 . 2 𝐴 / 𝑥𝐵 / 𝑦𝑦 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑥𝐶
84, 6, 73eqtr3g 2788 1 (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑥𝐶 = 𝐴 / 𝑥𝐵 / 𝑥𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1531   = wceq 1533  wcel 2098  wnfc 2875  csb 3889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-v 3463  df-sbc 3774  df-csb 3890
This theorem is referenced by:  csbidm  4432
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