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Theorem cdleme3d 39736
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 39741 and cdleme3 39742. (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 7437 . . 3 (𝑄 ∨ 𝑉) = (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ π‘Š))
43oveq2i 7437 . 2 ((𝑅 ∨ π‘ˆ) ∧ (𝑄 ∨ 𝑉)) = ((𝑅 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ π‘Š)))
51, 4eqtr4i 2759 1 𝐹 = ((𝑅 ∨ π‘ˆ) ∧ (𝑄 ∨ 𝑉))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  β€˜cfv 6553  (class class class)co 7426  lecple 17247  joincjn 18310  meetcmee 18311  Atomscatm 38767  LHypclh 39489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-ov 7429
This theorem is referenced by:  cdleme3g  39739  cdleme3h  39740  cdleme9  39758
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