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	$⊢ χ \leftrightarrow (n \in D \land f Fn n \land φ \land ψ)$
	bnj984.11	$⊢ B = \{f \| \exists n \in D (f Fn n \land φ \land ψ)\}$
Assertion	bnj984	$⊢ G \in A \to (G \in B \leftrightarrow \underset{˙}{[} G / f \underset{˙}{]} \exists n χ)$

Proof

Step	Hyp	Ref	Expression
1		bnj984.3	$⊢ χ \leftrightarrow (n \in D \land f Fn n \land φ \land ψ)$
2		bnj984.11	$⊢ B = \{f \| \exists n \in D (f Fn n \land φ \land ψ)\}$
3	2	eleq2i	$⊢ G \in B \leftrightarrow G \in \{f \| \exists n \in D (f Fn n \land φ \land ψ)\}$
4		sbc8g	$⊢ G \in A \to (\underset{˙}{[} G / f \underset{˙}{]} \exists n \in D (f Fn n \land φ \land ψ) \leftrightarrow G \in \{f \| \exists n \in D (f Fn n \land φ \land ψ)\})$
5	3 4	bitr4id	$⊢ G \in A \to (G \in B \leftrightarrow \underset{˙}{[} G / f \underset{˙}{]} \exists n \in D (f Fn n \land φ \land ψ))$
6		df-rex	$⊢ \exists n \in D (f Fn n \land φ \land ψ) \leftrightarrow \exists n (n \in D \land (f Fn n \land φ \land ψ))$
7		bnj252	$⊢ (n \in D \land f Fn n \land φ \land ψ) \leftrightarrow (n \in D \land (f Fn n \land φ \land ψ))$
8	1 7	bitri	$⊢ χ \leftrightarrow (n \in D \land (f Fn n \land φ \land ψ))$
9	6 8	bnj133	$⊢ \exists n \in D (f Fn n \land φ \land ψ) \leftrightarrow \exists n χ$
10	9	sbcbii	$⊢ \underset{˙}{[} G / f \underset{˙}{]} \exists n \in D (f Fn n \land φ \land ψ) \leftrightarrow \underset{˙}{[} G / f \underset{˙}{]} \exists n χ$
11	5 10	bitrdi	$⊢ G \in A \to (G \in B \leftrightarrow \underset{˙}{[} G / f \underset{˙}{]} \exists n χ)$

Metamath Proof Explorer

Theorem bnj984

Proof