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Theorem cdlemk40f 38127
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
cdlemk40f ((𝐹𝑁𝐺𝑇) → (𝑈𝐺) = 𝐺 / 𝑔𝑋)
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 cdlemk40f
StepHypRef Expression
1 cdlemk40.x . . 3 𝑋 = (𝑧𝑇 𝜑)
2 cdlemk40.u . . 3 𝑈 = (𝑔𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))
31, 2cdlemk40 38125 . 2 (𝐺𝑇 → (𝑈𝐺) = if(𝐹 = 𝑁, 𝐺, 𝐺 / 𝑔𝑋))
4 ifnefalse 4462 . 2 (𝐹𝑁 → if(𝐹 = 𝑁, 𝐺, 𝐺 / 𝑔𝑋) = 𝐺 / 𝑔𝑋)
53, 4sylan9eqr 2881 1 ((𝐹𝑁𝐺𝑇) → (𝑈𝐺) = 𝐺 / 𝑔𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wne 3014  csb 3866  ifcif 4450  cmpt 5133  cfv 6344  crio 7103
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 5190  ax-nul 5197  ax-pr 5318
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-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-iota 6303  df-fun 6346  df-fv 6352  df-riota 7104
This theorem is referenced by:  cdlemk43N  38171  cdlemk35u  38172  cdlemk55u1  38173  cdlemk39u1  38175  cdlemk19u1  38177
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