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Theorem cdlemk40t 37934
Description: TODO: fix comment. (Contributed by NM, 31-Jul-2013.)
Hypotheses
Ref Expression
cdlemk40.x 𝑋 = (𝑧𝑇 𝜑)
cdlemk40.u 𝑈 = (𝑔𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))
Assertion
Ref Expression
cdlemk40t ((𝐹 = 𝑁𝐺𝑇) → (𝑈𝐺) = 𝐺)
Distinct variable groups:   𝑔,𝐹   𝑔,𝑁   𝑇,𝑔
Allowed substitution hints:   𝜑(𝑧,𝑔)   𝑇(𝑧)   𝑈(𝑧,𝑔)   𝐹(𝑧)   𝐺(𝑧,𝑔)   𝑁(𝑧)   𝑋(𝑧,𝑔)

Proof of Theorem cdlemk40t
StepHypRef Expression
1 cdlemk40.x . . 3 𝑋 = (𝑧𝑇 𝜑)
2 cdlemk40.u . . 3 𝑈 = (𝑔𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))
31, 2cdlemk40 37933 . 2 (𝐺𝑇 → (𝑈𝐺) = if(𝐹 = 𝑁, 𝐺, 𝐺 / 𝑔𝑋))
4 iftrue 4469 . 2 (𝐹 = 𝑁 → if(𝐹 = 𝑁, 𝐺, 𝐺 / 𝑔𝑋) = 𝐺)
53, 4sylan9eqr 2875 1 ((𝐹 = 𝑁𝐺𝑇) → (𝑈𝐺) = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  csb 3880  ifcif 4463  cmpt 5137  cfv 6348  crio 7102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-riota 7103
This theorem is referenced by:  cdlemk35u  37980  cdlemk55u  37982  cdlemk39u  37984  cdlemk19u  37986
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