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Theorem cdleme31sdnN 36461
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 253 . . 3 (𝑠 (𝑃 𝑄) ↔ 𝑠 (𝑃 𝑄))
3 vex 3417 . . . 4 𝑠 ∈ V
4 cdleme31sdn.d . . . . 5 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
5 cdleme31sdn.c . . . . 5 𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
64, 5cdleme31sc 36458 . . . 4 (𝑠 ∈ V → 𝑠 / 𝑡𝐷 = 𝐶)
73, 6ax-mp 5 . . 3 𝑠 / 𝑡𝐷 = 𝐶
82, 7ifbieq2i 4332 . 2 if(𝑠 (𝑃 𝑄), 𝐼, 𝑠 / 𝑡𝐷) = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)
91, 8eqtr4i 2852 1 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝑠 / 𝑡𝐷)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1656  wcel 2164  Vcvv 3414  csb 3757  ifcif 4308   class class class wbr 4875  (class class class)co 6910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-iota 6090  df-fv 6135  df-ov 6913
This theorem is referenced by: (None)
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