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Metamath Proof Explorer

Theorem bnj1345

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj1345.1	$⊢ φ \to \exists x (ψ \land χ)$
	bnj1345.2	$⊢ θ \leftrightarrow (φ \land ψ \land χ)$
	bnj1345.3	$⊢ φ \to \forall x φ$
Assertion	bnj1345	$⊢ φ \to \exists x θ$

Step	Hyp	Ref	Expression
1		bnj1345.1	$⊢ φ \to \exists x (ψ \land χ)$
2		bnj1345.2	$⊢ θ \leftrightarrow (φ \land ψ \land χ)$
3		bnj1345.3	$⊢ φ \to \forall x φ$
4	1 3	bnj1275	$⊢ φ \to \exists x (φ \land ψ \land χ)$
5	4 2	bnj1198	$⊢ φ \to \exists x θ$