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Theorem cdleme31sdnN 40389
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 40386 . . . 4 (𝑠 ∈ V → 𝑠 / 𝑡𝐷 = 𝐶)
65elv 3485 . . 3 𝑠 / 𝑡𝐷 = 𝐶
72, 6ifbieq2i 4551 . 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 1540  Vcvv 3480  csb 3899  ifcif 4525   class class class wbr 5143  (class class class)co 7431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434
This theorem is referenced by: (None)
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