bnj1083

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	bnj1083.3	$⊢ χ \leftrightarrow (n \in D \land f Fn n \land φ \land ψ)$
	bnj1083.8	$⊢ K = \{f \| \exists n \in D (f Fn n \land φ \land ψ)\}$
Assertion	bnj1083	$⊢ f \in K \leftrightarrow \exists n χ$

Proof

Step	Hyp	Ref	Expression
1		bnj1083.3	$⊢ χ \leftrightarrow (n \in D \land f Fn n \land φ \land ψ)$
2		bnj1083.8	$⊢ K = \{f \| \exists n \in D (f Fn n \land φ \land ψ)\}$
3		df-rex	$⊢ \exists n \in D (f Fn n \land φ \land ψ) \leftrightarrow \exists n (n \in D \land (f Fn n \land φ \land ψ))$
4	2	abeq2i	$⊢ f \in K \leftrightarrow \exists n \in D (f Fn n \land φ \land ψ)$
5		bnj252	$⊢ (n \in D \land f Fn n \land φ \land ψ) \leftrightarrow (n \in D \land (f Fn n \land φ \land ψ))$
6	1 5	bitri	$⊢ χ \leftrightarrow (n \in D \land (f Fn n \land φ \land ψ))$
7	6	exbii	$⊢ \exists n χ \leftrightarrow \exists n (n \in D \land (f Fn n \land φ \land ψ))$
8	3 4 7	3bitr4i	$⊢ f \in K \leftrightarrow \exists n χ$

Metamath Proof Explorer

Theorem bnj1083

Proof