Theorem cdleme3d 40816

Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 40821 and cdleme3 40822. (Contributed by NM, 6-Jun-2012.)

Hypotheses

Ref	Expression
cdleme1.l	⊢ ≤ = (le‘𝐾)
cdleme1.j	⊢ ∨ = (join‘𝐾)
cdleme1.m	⊢ ∧ = (meet‘𝐾)
cdleme1.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme1.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme1.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme1.f	⊢ 𝐹 = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)))
cdleme3.3	⊢ 𝑉 = ((𝑃 ∨ 𝑅) ∧ 𝑊)

Assertion

Ref	Expression
cdleme3d	⊢ 𝐹 = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ 𝑉))

Proof of Theorem cdleme3d

Step	Hyp	Ref	Expression
1		cdleme1.f	. 2 ⊢ 𝐹 = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)))
2		cdleme3.3	. . . 4 ⊢ 𝑉 = ((𝑃 ∨ 𝑅) ∧ 𝑊)
3	2	oveq2i 7402	. . 3 ⊢ (𝑄 ∨ 𝑉) = (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))
4	3	oveq2i 7402	. 2 ⊢ ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ 𝑉)) = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)))
5	1, 4	eqtr4i 2787	1 ⊢ 𝐹 = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ 𝑉))

Syntax hints:   = wceq 1559  ‘cfv 6516  (class class class)co 7391  lecple 17284  joincjn 18334  meetcmee 18335  Atomscatm 39848  LHypclh 40569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-iota 6472  df-fv 6524  df-ov 7394

This theorem is referenced by:  cdleme3g  40819  cdleme3h  40820  cdleme9  40838