Theorem cdleme7a 40220

Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 40225 and cdleme7 40226. (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 7424	. . 3 ⊢ (𝐹 ∨ 𝑉) = (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊))
4	3	oveq2i 7424	. 2 ⊢ ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝑉)) = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)))
5	1, 4	eqtr4i 2760	1 ⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝑉))

Syntax hints:   = wceq 1539  ‘cfv 6541  (class class class)co 7413  lecple 17281  joincjn 18328  meetcmee 18329  Atomscatm 39239  LHypclh 39961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-iota 6494  df-fv 6549  df-ov 7416

This theorem is referenced by:  cdleme7d  40223  cdleme17a  40263