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Metamath Proof Explorer

Theorem bnj984

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 bnj984.3 ⊢ χ ↔ n ∈ D ∧ f Fn n ∧ φ ∧ ψ
bnj984.11 ⊢ B = f | ∃ n ∈ D f Fn n ∧ φ ∧ ψ
Assertion bnj984 ⊢ G ∈ A → G ∈ B ↔ [˙G / f]˙ ∃ n χ

Proof

Step Hyp Ref Expression
1 bnj984.3 ⊢ χ ↔ n ∈ D ∧ f Fn n ∧ φ ∧ ψ
2 bnj984.11 ⊢ B = f | ∃ n ∈ D f Fn n ∧ φ ∧ ψ
3 2 eleq2i ⊢ G ∈ B ↔ G ∈ f | ∃ n ∈ D f Fn n ∧ φ ∧ ψ
4 sbc8g ⊢ G ∈ A → [˙G / f]˙ ∃ n ∈ D f Fn n ∧ φ ∧ ψ ↔ G ∈ f | ∃ n ∈ D f Fn n ∧ φ ∧ ψ
5 3 4 bitr4id ⊢ G ∈ A → G ∈ B ↔ [˙G / f]˙ ∃ n ∈ D f Fn n ∧ φ ∧ ψ
6 df-rex ⊢ ∃ n ∈ D f Fn n ∧ φ ∧ ψ ↔ ∃ n n ∈ D ∧ f Fn n ∧ φ ∧ ψ
7 bnj252 ⊢ n ∈ D ∧ f Fn n ∧ φ ∧ ψ ↔ n ∈ D ∧ f Fn n ∧ φ ∧ ψ
8 1 7 bitri ⊢ χ ↔ n ∈ D ∧ f Fn n ∧ φ ∧ ψ
9 6 8 bnj133 ⊢ ∃ n ∈ D f Fn n ∧ φ ∧ ψ ↔ ∃ n χ
10 9 sbcbii ⊢ [˙G / f]˙ ∃ n ∈ D f Fn n ∧ φ ∧ ψ ↔ [˙G / f]˙ ∃ n χ
11 5 10 bitrdi ⊢ G ∈ A → G ∈ B ↔ [˙G / f]˙ ∃ n χ