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Theorem csbmpt12 5515
Description: Move substitution into a maps-to notation. (Contributed by AV, 26-Sep-2019.)
Assertion
Ref Expression
csbmpt12 (𝐴𝑉𝐴 / 𝑥(𝑦𝑌𝑍) = (𝑦𝐴 / 𝑥𝑌𝐴 / 𝑥𝑍))
Distinct variable groups:   𝑦,𝐴   𝑦,𝑉   𝑦,𝑌   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥)   𝑉(𝑥)   𝑌(𝑥)   𝑍(𝑥,𝑦)

Proof of Theorem csbmpt12
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 csbopab 5513 . . 3 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ (𝑦𝑌𝑧 = 𝑍)} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥](𝑦𝑌𝑧 = 𝑍)}
2 sbcan 3792 . . . . 5 ([𝐴 / 𝑥](𝑦𝑌𝑧 = 𝑍) ↔ ([𝐴 / 𝑥]𝑦𝑌[𝐴 / 𝑥]𝑧 = 𝑍))
3 sbcel12 4369 . . . . . . 7 ([𝐴 / 𝑥]𝑦𝑌𝐴 / 𝑥𝑦𝐴 / 𝑥𝑌)
4 csbconstg 3875 . . . . . . . 8 (𝐴𝑉𝐴 / 𝑥𝑦 = 𝑦)
54eleq1d 2823 . . . . . . 7 (𝐴𝑉 → (𝐴 / 𝑥𝑦𝐴 / 𝑥𝑌𝑦𝐴 / 𝑥𝑌))
63, 5bitrid 283 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝑌𝑦𝐴 / 𝑥𝑌))
7 sbceq2g 4377 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧 = 𝑍𝑧 = 𝐴 / 𝑥𝑍))
86, 7anbi12d 632 . . . . 5 (𝐴𝑉 → (([𝐴 / 𝑥]𝑦𝑌[𝐴 / 𝑥]𝑧 = 𝑍) ↔ (𝑦𝐴 / 𝑥𝑌𝑧 = 𝐴 / 𝑥𝑍)))
92, 8bitrid 283 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥](𝑦𝑌𝑧 = 𝑍) ↔ (𝑦𝐴 / 𝑥𝑌𝑧 = 𝐴 / 𝑥𝑍)))
109opabbidv 5172 . . 3 (𝐴𝑉 → {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥](𝑦𝑌𝑧 = 𝑍)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴 / 𝑥𝑌𝑧 = 𝐴 / 𝑥𝑍)})
111, 10eqtrid 2789 . 2 (𝐴𝑉𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ (𝑦𝑌𝑧 = 𝑍)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴 / 𝑥𝑌𝑧 = 𝐴 / 𝑥𝑍)})
12 df-mpt 5190 . . 3 (𝑦𝑌𝑍) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝑌𝑧 = 𝑍)}
1312csbeq2i 3864 . 2 𝐴 / 𝑥(𝑦𝑌𝑍) = 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ (𝑦𝑌𝑧 = 𝑍)}
14 df-mpt 5190 . 2 (𝑦𝐴 / 𝑥𝑌𝐴 / 𝑥𝑍) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴 / 𝑥𝑌𝑧 = 𝐴 / 𝑥𝑍)}
1511, 13, 143eqtr4g 2802 1 (𝐴𝑉𝐴 / 𝑥(𝑦𝑌𝑍) = (𝑦𝐴 / 𝑥𝑌𝐴 / 𝑥𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  [wsbc 3740  csb 3856  {copab 5168  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:  csbmpt2  5516  esum2dlem  32694
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