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Theorem csbmpt2 5153
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 5152 . 2 (𝐴𝑉𝐴 / 𝑥(𝑦𝑌𝑍) = (𝑦𝐴 / 𝑥𝑌𝐴 / 𝑥𝑍))
2 csbconstg 3679 . . 3 (𝐴𝑉𝐴 / 𝑥𝑌 = 𝑌)
32mpteq1d 4882 . 2 (𝐴𝑉 → (𝑦𝐴 / 𝑥𝑌𝐴 / 𝑥𝑍) = (𝑦𝑌𝐴 / 𝑥𝑍))
41, 3eqtrd 2786 1 (𝐴𝑉𝐴 / 𝑥(𝑦𝑌𝑍) = (𝑦𝑌𝐴 / 𝑥𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1624  wcel 2131  csb 3666  cmpt 4873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-fal 1630  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-opab 4857  df-mpt 4874
This theorem is referenced by:  matgsum  20437  csbrdgg  33478
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