Theorem cdleme3d 40730

Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 40735 and cdleme3 40736. (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 7374	. . 3 ⊢ (𝑄 ∨ 𝑉) = (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))
4	3	oveq2i 7374	. 2 ⊢ ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ 𝑉)) = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)))
5	1, 4	eqtr4i 2766	1 ⊢ 𝐹 = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ 𝑉))

Syntax hints:   = wceq 1547  ‘cfv 6492  (class class class)co 7363  lecple 17225  joincjn 18275  meetcmee 18276  Atomscatm 39762  LHypclh 40483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-iota 6448  df-fv 6500  df-ov 7366

This theorem is referenced by:  cdleme3g  40733  cdleme3h  40734  cdleme9  40752