Metamath Proof Explorer

Theorem cdleme28c

Description: Part of proof of Lemma E in Crawley p. 113. Eliminate the s =/= t antecedent in cdleme28b . TODO: FIX COMMENT. (Contributed by NM, 6-Feb-2013)

Ref Expression
Hypotheses cdleme26.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdleme26.l ⊢ ≤ = ( le ‘ 𝐾 )
cdleme26.j ⊢ ∨ = ( join ‘ 𝐾 )
cdleme26.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdleme26.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdleme26.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdleme27.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
cdleme27.f ⊢ 𝐹 = ( ( 𝑠 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) ) )
cdleme27.z ⊢ 𝑍 = ( ( 𝑧 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑧 ) ∧ 𝑊 ) ) )
cdleme27.n ⊢ 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑠 ∨ 𝑧 ) ∧ 𝑊 ) ) )
cdleme27.d ⊢ 𝐷 = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 ) )
cdleme27.c ⊢ 𝐶 = if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐷 , 𝐹 )
cdleme27.g ⊢ 𝐺 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) )
cdleme27.o ⊢ 𝑂 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑡 ∨ 𝑧 ) ∧ 𝑊 ) ) )
cdleme27.e ⊢ 𝐸 = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑂 ) )
cdleme27.y ⊢ 𝑌 = if ( 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐸 , 𝐺 )
Assertion cdleme28c ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ) → ( 𝐶 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑌 ∨ ( 𝑋 ∧ 𝑊 ) ) )

Proof

Step Hyp Ref Expression
1 cdleme26.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdleme26.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdleme26.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cdleme26.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 cdleme26.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 cdleme26.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 cdleme27.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
8 cdleme27.f ⊢ 𝐹 = ( ( 𝑠 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) ) )
9 cdleme27.z ⊢ 𝑍 = ( ( 𝑧 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑧 ) ∧ 𝑊 ) ) )
10 cdleme27.n ⊢ 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑠 ∨ 𝑧 ) ∧ 𝑊 ) ) )
11 cdleme27.d ⊢ 𝐷 = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 ) )
12 cdleme27.c ⊢ 𝐶 = if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐷 , 𝐹 )
13 cdleme27.g ⊢ 𝐺 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) )
14 cdleme27.o ⊢ 𝑂 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑡 ∨ 𝑧 ) ∧ 𝑊 ) ) )
15 cdleme27.e ⊢ 𝐸 = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑂 ) )
16 cdleme27.y ⊢ 𝑌 = if ( 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐸 , 𝐺 )
17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 cdleme27b ⊢ ( 𝑠 = 𝑡 → 𝐶 = 𝑌 )
18 17 oveq1d ⊢ ( 𝑠 = 𝑡 → ( 𝐶 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑌 ∨ ( 𝑋 ∧ 𝑊 ) ) )
19 18 adantl ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ) ∧ 𝑠 = 𝑡 ) → ( 𝐶 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑌 ∨ ( 𝑋 ∧ 𝑊 ) ) )
20 simpl11 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ) ∧ 𝑠 ≠ 𝑡 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
21 simpl12 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ) ∧ 𝑠 ≠ 𝑡 ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
22 simpl13 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ) ∧ 𝑠 ≠ 𝑡 ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
23 simpl21 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ) ∧ 𝑠 ≠ 𝑡 ) → 𝑃 ≠ 𝑄 )
24 simpl22 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ) ∧ 𝑠 ≠ 𝑡 ) → ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) )
25 simpl23 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ) ∧ 𝑠 ≠ 𝑡 ) → ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) )
26 simpr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ) ∧ 𝑠 ≠ 𝑡 ) → 𝑠 ≠ 𝑡 )
27 simpl31 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ) ∧ 𝑠 ≠ 𝑡 ) → ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 )
28 simpl32 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ) ∧ 𝑠 ≠ 𝑡 ) → ( 𝑡 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 )
29 27 28 jca ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ) ∧ 𝑠 ≠ 𝑡 ) → ( ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) )
30 simpl33 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ) ∧ 𝑠 ≠ 𝑡 ) → ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) )
31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 cdleme28b ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( 𝑠 ≠ 𝑡 ∧ ( ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ) → ( 𝐶 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑌 ∨ ( 𝑋 ∧ 𝑊 ) ) )
32 20 21 22 23 24 25 26 29 30 31 syl333anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ) ∧ 𝑠 ≠ 𝑡 ) → ( 𝐶 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑌 ∨ ( 𝑋 ∧ 𝑊 ) ) )
33 19 32 pm2.61dane ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑡 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) ) → ( 𝐶 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑌 ∨ ( 𝑋 ∧ 𝑊 ) ) )