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Metamath Proof Explorer

Theorem bnj1171

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 bnj1171.13 ⊢ φ ∧ ψ → B ⊆ A
bnj1171.129 ⊢ ∃ z ∀ w φ ∧ ψ → z ∈ B ∧ w ∈ A → w R z → ¬ w ∈ B
Assertion bnj1171 ⊢ ∃ z ∀ w φ ∧ ψ → z ∈ B ∧ w ∈ B → ¬ w R z

Proof

Step Hyp Ref Expression
1 bnj1171.13 ⊢ φ ∧ ψ → B ⊆ A
2 bnj1171.129 ⊢ ∃ z ∀ w φ ∧ ψ → z ∈ B ∧ w ∈ A → w R z → ¬ w ∈ B
3 1 sseld ⊢ φ ∧ ψ → w ∈ B → w ∈ A
4 3 pm4.71rd ⊢ φ ∧ ψ → w ∈ B ↔ w ∈ A ∧ w ∈ B
5 4 imbi1d ⊢ φ ∧ ψ → w ∈ B → ¬ w R z ↔ w ∈ A ∧ w ∈ B → ¬ w R z
6 impexp ⊢ w ∈ A ∧ w ∈ B → ¬ w R z ↔ w ∈ A → w ∈ B → ¬ w R z
7 5 6 bitrdi ⊢ φ ∧ ψ → w ∈ B → ¬ w R z ↔ w ∈ A → w ∈ B → ¬ w R z
8 con2b ⊢ w R z → ¬ w ∈ B ↔ w ∈ B → ¬ w R z
9 8 imbi2i ⊢ w ∈ A → w R z → ¬ w ∈ B ↔ w ∈ A → w ∈ B → ¬ w R z
10 7 9 bitr4di ⊢ φ ∧ ψ → w ∈ B → ¬ w R z ↔ w ∈ A → w R z → ¬ w ∈ B
11 10 anbi2d ⊢ φ ∧ ψ → z ∈ B ∧ w ∈ B → ¬ w R z ↔ z ∈ B ∧ w ∈ A → w R z → ¬ w ∈ B
12 11 pm5.74i ⊢ φ ∧ ψ → z ∈ B ∧ w ∈ B → ¬ w R z ↔ φ ∧ ψ → z ∈ B ∧ w ∈ A → w R z → ¬ w ∈ B
13 12 albii ⊢ ∀ w φ ∧ ψ → z ∈ B ∧ w ∈ B → ¬ w R z ↔ ∀ w φ ∧ ψ → z ∈ B ∧ w ∈ A → w R z → ¬ w ∈ B
14 13 exbii ⊢ ∃ z ∀ w φ ∧ ψ → z ∈ B ∧ w ∈ B → ¬ w R z ↔ ∃ z ∀ w φ ∧ ψ → z ∈ B ∧ w ∈ A → w R z → ¬ w ∈ B
15 2 14 mpbir ⊢ ∃ z ∀ w φ ∧ ψ → z ∈ B ∧ w ∈ B → ¬ w R z