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Theorem csbmpt2 5516
Description: Move substitution into the second part of a maps-to notation. (Contributed by AV, 26-Sep-2019.)
Assertion
Ref Expression
csbmpt2 (𝐴𝑉𝐴 / 𝑥(𝑦𝑌𝑍) = (𝑦𝑌𝐴 / 𝑥𝑍))
Distinct variable groups:   𝑦,𝐴   𝑦,𝑉   𝑦,𝑌,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝑉(𝑥)   𝑍(𝑥,𝑦)

Proof of Theorem csbmpt2
StepHypRef Expression
1 csbmpt12 5515 . 2 (𝐴𝑉𝐴 / 𝑥(𝑦𝑌𝑍) = (𝑦𝐴 / 𝑥𝑌𝐴 / 𝑥𝑍))
2 csbconstg 3875 . . 3 (𝐴𝑉𝐴 / 𝑥𝑌 = 𝑌)
32mpteq1d 5201 . 2 (𝐴𝑉 → (𝑦𝐴 / 𝑥𝑌𝐴 / 𝑥𝑍) = (𝑦𝑌𝐴 / 𝑥𝑍))
41, 3eqtrd 2777 1 (𝐴𝑉𝐴 / 𝑥(𝑦𝑌𝑍) = (𝑦𝑌𝐴 / 𝑥𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  csb 3856  cmpt 5189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-opab 5169  df-mpt 5190
This theorem is referenced by:  matgsum  21789  csbrdgg  35803
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