Theorem bnj1186 32887

Description: Technical lemma for bnj69 32890. 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.)

Hypothesis

Ref	Expression
bnj1186.1	⊢ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧)))

Assertion

Ref	Expression
bnj1186	⊢ ((𝜑 ∧ 𝜓) → ∃𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ¬ 𝑤𝑅𝑧)

Distinct variable groups:   𝑤,𝐵   𝜑,𝑤,𝑧   𝜓,𝑤,𝑧

Allowed substitution hints:   𝐵(𝑧)   𝑅(𝑧,𝑤)

Proof of Theorem bnj1186

Step	Hyp	Ref	Expression
1		bnj1186.1	. . . . . 6 ⊢ ∃𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧)))
2		19.21v 1943	. . . . . . 7 ⊢ (∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧))) ↔ ((𝜑 ∧ 𝜓) → ∀𝑤(𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧))))
3	2	exbii 1851	. . . . . 6 ⊢ (∃𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧))) ↔ ∃𝑧((𝜑 ∧ 𝜓) → ∀𝑤(𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧))))
4	1, 3	mpbi 229	. . . . 5 ⊢ ∃𝑧((𝜑 ∧ 𝜓) → ∀𝑤(𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧)))
5	4	19.37iv 1953	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑧∀𝑤(𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧)))
6		19.28v 1995	. . . . 5 ⊢ (∀𝑤(𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧)) ↔ (𝑧 ∈ 𝐵 ∧ ∀𝑤(𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧)))
7	6	exbii 1851	. . . 4 ⊢ (∃𝑧∀𝑤(𝑧 ∈ 𝐵 ∧ (𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧)) ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ ∀𝑤(𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧)))
8	5, 7	sylib 217	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑧(𝑧 ∈ 𝐵 ∧ ∀𝑤(𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧)))
9		df-ral 3068	. . . . 5 ⊢ (∀𝑤 ∈ 𝐵 ¬ 𝑤𝑅𝑧 ↔ ∀𝑤(𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧))
10	9	anbi2i 622	. . . 4 ⊢ ((𝑧 ∈ 𝐵 ∧ ∀𝑤 ∈ 𝐵 ¬ 𝑤𝑅𝑧) ↔ (𝑧 ∈ 𝐵 ∧ ∀𝑤(𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧)))
11	10	exbii 1851	. . 3 ⊢ (∃𝑧(𝑧 ∈ 𝐵 ∧ ∀𝑤 ∈ 𝐵 ¬ 𝑤𝑅𝑧) ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ ∀𝑤(𝑤 ∈ 𝐵 → ¬ 𝑤𝑅𝑧)))
12	8, 11	sylibr 233	. 2 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑧(𝑧 ∈ 𝐵 ∧ ∀𝑤 ∈ 𝐵 ¬ 𝑤𝑅𝑧))
13		df-rex 3069	. 2 ⊢ (∃𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ¬ 𝑤𝑅𝑧 ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ ∀𝑤 ∈ 𝐵 ¬ 𝑤𝑅𝑧))
14	12, 13	sylibr 233	1 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ¬ 𝑤𝑅𝑧)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1537  ∃wex 1783   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   class class class wbr 5070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-ral 3068  df-rex 3069

This theorem is referenced by:  bnj1190  32888