Theorem cdleme7a 40619

Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 40624 and cdleme7 40625. (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

Step	Hyp	Ref	Expression
1		cdleme4.g	. 2 ⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)))
2		cdleme7.v	. . . 4 ⊢ 𝑉 = ((𝑅 ∨ 𝑆) ∧ 𝑊)
3	2	oveq2i 7379	. . 3 ⊢ (𝐹 ∨ 𝑉) = (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊))
4	3	oveq2i 7379	. 2 ⊢ ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝑉)) = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)))
5	1, 4	eqtr4i 2763	1 ⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝑉))

Syntax hints:   = wceq 1542  ‘cfv 6500  (class class class)co 7368  lecple 17196  joincjn 18246  meetcmee 18247  Atomscatm 39639  LHypclh 40360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371

This theorem is referenced by:  cdleme7d  40622  cdleme17a  40662