Metamath Proof Explorer

Theorem bnj1186

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
Hypothesis bnj1186.1 ⊢ ∃ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑧 ∈ 𝐵 ∧ ( 𝑤 ∈ 𝐵 → ¬ 𝑤 𝑅 𝑧 ) ) )
Assertion bnj1186 ( ( 𝜑 ∧ 𝜓 ) → ∃ 𝑧 ∈ 𝐵 ∀ 𝑤 ∈ 𝐵 ¬ 𝑤 𝑅 𝑧 )

Proof

Step Hyp Ref Expression
1 bnj1186.1 ⊢ ∃ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑧 ∈ 𝐵 ∧ ( 𝑤 ∈ 𝐵 → ¬ 𝑤 𝑅 𝑧 ) ) )
2 19.21v ⊢ ( ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑧 ∈ 𝐵 ∧ ( 𝑤 ∈ 𝐵 → ¬ 𝑤 𝑅 𝑧 ) ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) → ∀ 𝑤 ( 𝑧 ∈ 𝐵 ∧ ( 𝑤 ∈ 𝐵 → ¬ 𝑤 𝑅 𝑧 ) ) ) )
3 2 exbii ⊢ ( ∃ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑧 ∈ 𝐵 ∧ ( 𝑤 ∈ 𝐵 → ¬ 𝑤 𝑅 𝑧 ) ) ) ↔ ∃ 𝑧 ( ( 𝜑 ∧ 𝜓 ) → ∀ 𝑤 ( 𝑧 ∈ 𝐵 ∧ ( 𝑤 ∈ 𝐵 → ¬ 𝑤 𝑅 𝑧 ) ) ) )
4 1 3 mpbi ⊢ ∃ 𝑧 ( ( 𝜑 ∧ 𝜓 ) → ∀ 𝑤 ( 𝑧 ∈ 𝐵 ∧ ( 𝑤 ∈ 𝐵 → ¬ 𝑤 𝑅 𝑧 ) ) )
5 4 19.37iv ⊢ ( ( 𝜑 ∧ 𝜓 ) → ∃ 𝑧 ∀ 𝑤 ( 𝑧 ∈ 𝐵 ∧ ( 𝑤 ∈ 𝐵 → ¬ 𝑤 𝑅 𝑧 ) ) )
6 19.28v ⊢ ( ∀ 𝑤 ( 𝑧 ∈ 𝐵 ∧ ( 𝑤 ∈ 𝐵 → ¬ 𝑤 𝑅 𝑧 ) ) ↔ ( 𝑧 ∈ 𝐵 ∧ ∀ 𝑤 ( 𝑤 ∈ 𝐵 → ¬ 𝑤 𝑅 𝑧 ) ) )
7 6 exbii ⊢ ( ∃ 𝑧 ∀ 𝑤 ( 𝑧 ∈ 𝐵 ∧ ( 𝑤 ∈ 𝐵 → ¬ 𝑤 𝑅 𝑧 ) ) ↔ ∃ 𝑧 ( 𝑧 ∈ 𝐵 ∧ ∀ 𝑤 ( 𝑤 ∈ 𝐵 → ¬ 𝑤 𝑅 𝑧 ) ) )
8 5 7 sylib ⊢ ( ( 𝜑 ∧ 𝜓 ) → ∃ 𝑧 ( 𝑧 ∈ 𝐵 ∧ ∀ 𝑤 ( 𝑤 ∈ 𝐵 → ¬ 𝑤 𝑅 𝑧 ) ) )
9 df-ral ⊢ ( ∀ 𝑤 ∈ 𝐵 ¬ 𝑤 𝑅 𝑧 ↔ ∀ 𝑤 ( 𝑤 ∈ 𝐵 → ¬ 𝑤 𝑅 𝑧 ) )
10 9 anbi2i ⊢ ( ( 𝑧 ∈ 𝐵 ∧ ∀ 𝑤 ∈ 𝐵 ¬ 𝑤 𝑅 𝑧 ) ↔ ( 𝑧 ∈ 𝐵 ∧ ∀ 𝑤 ( 𝑤 ∈ 𝐵 → ¬ 𝑤 𝑅 𝑧 ) ) )
11 10 exbii ⊢ ( ∃ 𝑧 ( 𝑧 ∈ 𝐵 ∧ ∀ 𝑤 ∈ 𝐵 ¬ 𝑤 𝑅 𝑧 ) ↔ ∃ 𝑧 ( 𝑧 ∈ 𝐵 ∧ ∀ 𝑤 ( 𝑤 ∈ 𝐵 → ¬ 𝑤 𝑅 𝑧 ) ) )
12 8 11 sylibr ⊢ ( ( 𝜑 ∧ 𝜓 ) → ∃ 𝑧 ( 𝑧 ∈ 𝐵 ∧ ∀ 𝑤 ∈ 𝐵 ¬ 𝑤 𝑅 𝑧 ) )
13 df-rex ⊢ ( ∃ 𝑧 ∈ 𝐵 ∀ 𝑤 ∈ 𝐵 ¬ 𝑤 𝑅 𝑧 ↔ ∃ 𝑧 ( 𝑧 ∈ 𝐵 ∧ ∀ 𝑤 ∈ 𝐵 ¬ 𝑤 𝑅 𝑧 ) )
14 12 13 sylibr ⊢ ( ( 𝜑 ∧ 𝜓 ) → ∃ 𝑧 ∈ 𝐵 ∀ 𝑤 ∈ 𝐵 ¬ 𝑤 𝑅 𝑧 )