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Theorem cdlemk40t 37067
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 37066 . 2 (𝐺𝑇 → (𝑈𝐺) = if(𝐹 = 𝑁, 𝐺, 𝐺 / 𝑔𝑋))
4 iftrue 4312 . 2 (𝐹 = 𝑁 → if(𝐹 = 𝑁, 𝐺, 𝐺 / 𝑔𝑋) = 𝐺)
53, 4sylan9eqr 2835 1 ((𝐹 = 𝑁𝐺𝑇) → (𝑈𝐺) = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2106  csb 3750  ifcif 4306  cmpt 4965  cfv 6135  crio 6882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-riota 6883
This theorem is referenced by:  cdlemk35u  37113  cdlemk55u  37115  cdlemk39u  37117  cdlemk19u  37119
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