Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme31sdnN Structured version   Visualization version   GIF version

Theorem cdleme31sdnN 38879
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 31-Mar-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdleme31sdn.c 𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
cdleme31sdn.d 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
cdleme31sdn.n 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)
Assertion
Ref Expression
cdleme31sdnN 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝑠 / 𝑡𝐷)
Distinct variable groups:   𝑡,   𝑡,   𝑡,𝑃   𝑡,𝑄   𝑡,𝑈   𝑡,𝑊   𝑡,𝑠
Allowed substitution hints:   𝐶(𝑡,𝑠)   𝐷(𝑡,𝑠)   𝑃(𝑠)   𝑄(𝑠)   𝑈(𝑠)   𝐼(𝑡,𝑠)   (𝑠)   (𝑡,𝑠)   (𝑠)   𝑁(𝑡,𝑠)   𝑊(𝑠)

Proof of Theorem cdleme31sdnN
StepHypRef Expression
1 cdleme31sdn.n . 2 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)
2 biid 261 . . 3 (𝑠 (𝑃 𝑄) ↔ 𝑠 (𝑃 𝑄))
3 cdleme31sdn.d . . . . 5 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
4 cdleme31sdn.c . . . . 5 𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
53, 4cdleme31sc 38876 . . . 4 (𝑠 ∈ V → 𝑠 / 𝑡𝐷 = 𝐶)
65elv 3454 . . 3 𝑠 / 𝑡𝐷 = 𝐶
72, 6ifbieq2i 4516 . 2 if(𝑠 (𝑃 𝑄), 𝐼, 𝑠 / 𝑡𝐷) = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)
81, 7eqtr4i 2768 1 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝑠 / 𝑡𝐷)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  Vcvv 3448  csb 3860  ifcif 4491   class class class wbr 5110  (class class class)co 7362
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
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-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-iota 6453  df-fv 6509  df-ov 7365
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator