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Theorem cdleme7a 40245
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 40250 and cdleme7 40251. (Contributed by NM, 7-Jun-2012.)
Hypotheses
Ref Expression
cdleme4.l = (le‘𝐾)
cdleme4.j = (join‘𝐾)
cdleme4.m = (meet‘𝐾)
cdleme4.a 𝐴 = (Atoms‘𝐾)
cdleme4.h 𝐻 = (LHyp‘𝐾)
cdleme4.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme4.f 𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
cdleme4.g 𝐺 = ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊)))
cdleme7.v 𝑉 = ((𝑅 𝑆) 𝑊)
Assertion
Ref Expression
cdleme7a 𝐺 = ((𝑃 𝑄) (𝐹 𝑉))

Proof of Theorem cdleme7a
StepHypRef Expression
1 cdleme4.g . 2 𝐺 = ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊)))
2 cdleme7.v . . . 4 𝑉 = ((𝑅 𝑆) 𝑊)
32oveq2i 7442 . . 3 (𝐹 𝑉) = (𝐹 ((𝑅 𝑆) 𝑊))
43oveq2i 7442 . 2 ((𝑃 𝑄) (𝐹 𝑉)) = ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊)))
51, 4eqtr4i 2768 1 𝐺 = ((𝑃 𝑄) (𝐹 𝑉))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  cfv 6561  (class class class)co 7431  lecple 17304  joincjn 18357  meetcmee 18358  Atomscatm 39264  LHypclh 39986
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-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-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  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:  cdleme7d  40248  cdleme17a  40288
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