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Theorem cdleme3d 38697
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 38702 and cdleme3 38703. (Contributed by NM, 6-Jun-2012.)
Hypotheses
Ref Expression
cdleme1.l ≀ = (leβ€˜πΎ)
cdleme1.j ∨ = (joinβ€˜πΎ)
cdleme1.m ∧ = (meetβ€˜πΎ)
cdleme1.a 𝐴 = (Atomsβ€˜πΎ)
cdleme1.h 𝐻 = (LHypβ€˜πΎ)
cdleme1.u π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
cdleme1.f 𝐹 = ((𝑅 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ π‘Š)))
cdleme3.3 𝑉 = ((𝑃 ∨ 𝑅) ∧ π‘Š)
Assertion
Ref Expression
cdleme3d 𝐹 = ((𝑅 ∨ π‘ˆ) ∧ (𝑄 ∨ 𝑉))

Proof of Theorem cdleme3d
StepHypRef Expression
1 cdleme1.f . 2 𝐹 = ((𝑅 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ π‘Š)))
2 cdleme3.3 . . . 4 𝑉 = ((𝑃 ∨ 𝑅) ∧ π‘Š)
32oveq2i 7369 . . 3 (𝑄 ∨ 𝑉) = (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ π‘Š))
43oveq2i 7369 . 2 ((𝑅 ∨ π‘ˆ) ∧ (𝑄 ∨ 𝑉)) = ((𝑅 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ π‘Š)))
51, 4eqtr4i 2768 1 𝐹 = ((𝑅 ∨ π‘ˆ) ∧ (𝑄 ∨ 𝑉))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  β€˜cfv 6497  (class class class)co 7358  lecple 17141  joincjn 18201  meetcmee 18202  Atomscatm 37728  LHypclh 38450
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-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-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361
This theorem is referenced by:  cdleme3g  38700  cdleme3h  38701  cdleme9  38719
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