Metamath Proof Explorer

Theorem bnj1172

Description: Technical lemma for bnj69 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses bnj1172.3 ⊢ 𝐶 = ( trCl ( 𝑋 , 𝐴 , 𝑅 ) ∩ 𝐵 )
bnj1172.96 ⊢ ∃ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶 ) ∧ ( 𝜃 → ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ) ) )
bnj1172.113 ⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶 ) → ( 𝜃 ↔ 𝑤 ∈ 𝐴 ) )
Assertion bnj1172 ∃ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑧 ∈ 𝐵 ∧ ( 𝑤 ∈ 𝐴 → ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ) ) )

Proof

Step Hyp Ref Expression
1 bnj1172.3 ⊢ 𝐶 = ( trCl ( 𝑋 , 𝐴 , 𝑅 ) ∩ 𝐵 )
2 bnj1172.96 ⊢ ∃ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶 ) ∧ ( 𝜃 → ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ) ) )
3 bnj1172.113 ⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶 ) → ( 𝜃 ↔ 𝑤 ∈ 𝐴 ) )
4 3 imbi1d ⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶 ) → ( ( 𝜃 → ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ) ↔ ( 𝑤 ∈ 𝐴 → ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ) ) )
5 4 pm5.32i ⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶 ) ∧ ( 𝜃 → ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ) ) ↔ ( ( 𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶 ) ∧ ( 𝑤 ∈ 𝐴 → ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ) ) )
6 5 imbi2i ⊢ ( ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶 ) ∧ ( 𝜃 → ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ) ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶 ) ∧ ( 𝑤 ∈ 𝐴 → ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ) ) ) )
7 6 albii ⊢ ( ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶 ) ∧ ( 𝜃 → ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ) ) ) ↔ ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶 ) ∧ ( 𝑤 ∈ 𝐴 → ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ) ) ) )
8 7 exbii ⊢ ( ∃ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶 ) ∧ ( 𝜃 → ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ) ) ) ↔ ∃ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶 ) ∧ ( 𝑤 ∈ 𝐴 → ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ) ) ) )
9 2 8 mpbi ⊢ ∃ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶 ) ∧ ( 𝑤 ∈ 𝐴 → ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ) ) )
10 simp3 ⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶 ) → 𝑧 ∈ 𝐶 )
11 10 1 eleqtrdi ⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶 ) → 𝑧 ∈ ( trCl ( 𝑋 , 𝐴 , 𝑅 ) ∩ 𝐵 ) )
12 11 elin2d ⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶 ) → 𝑧 ∈ 𝐵 )
13 12 anim1i ⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶 ) ∧ ( 𝑤 ∈ 𝐴 → ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ) ) → ( 𝑧 ∈ 𝐵 ∧ ( 𝑤 ∈ 𝐴 → ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ) ) )
14 13 imim2i ⊢ ( ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶 ) ∧ ( 𝑤 ∈ 𝐴 → ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ) ) ) → ( ( 𝜑 ∧ 𝜓 ) → ( 𝑧 ∈ 𝐵 ∧ ( 𝑤 ∈ 𝐴 → ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ) ) ) )
15 14 alimi ⊢ ( ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶 ) ∧ ( 𝑤 ∈ 𝐴 → ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ) ) ) → ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑧 ∈ 𝐵 ∧ ( 𝑤 ∈ 𝐴 → ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ) ) ) )
16 9 15 bnj101 ⊢ ∃ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑧 ∈ 𝐵 ∧ ( 𝑤 ∈ 𝐴 → ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ) ) )