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Theorem csbmpt2 5432
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 5431 . 2 (𝐴𝑉𝐴 / 𝑥(𝑦𝑌𝑍) = (𝑦𝐴 / 𝑥𝑌𝐴 / 𝑥𝑍))
2 csbconstg 3885 . . 3 (𝐴𝑉𝐴 / 𝑥𝑌 = 𝑌)
32mpteq1d 5141 . 2 (𝐴𝑉 → (𝑦𝐴 / 𝑥𝑌𝐴 / 𝑥𝑍) = (𝑦𝑌𝐴 / 𝑥𝑍))
41, 3eqtrd 2859 1 (𝐴𝑉𝐴 / 𝑥(𝑦𝑌𝑍) = (𝑦𝑌𝐴 / 𝑥𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  csb 3866  cmpt 5132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-sn 4550  df-pr 4552  df-op 4556  df-opab 5115  df-mpt 5133
This theorem is referenced by:  matgsum  21039  csbrdgg  34658
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