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∀wφ∧ψ→z∈B∧w∈A→wRz→¬w∈B
Assertion bnj1171 ⊢∃z∀wφ∧ψ→z∈B∧w∈B→¬wRz

Proof

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