Metamath Proof Explorer

Theorem 19.40b

Description: The antecedent provides a condition implying the converse of 19.40 . This is to 19.40 what 19.33b is to 19.33 . (Contributed by BJ, 6-May-2019) (Proof shortened by Wolf Lammen, 13-Nov-2020)

		Ref	Expression
	Assertion	19.40b	$⊢ (\forall x φ \lor \forall x ψ) \to ((\exists x φ \land \exists x ψ) \leftrightarrow \exists x (φ \land ψ))$

Proof

Step	Hyp	Ref	Expression
1		pm3.21	$⊢ ψ \to (φ \to (φ \land ψ))$
2	1	aleximi	$⊢ \forall x ψ \to (\exists x φ \to \exists x (φ \land ψ))$
3		pm3.2	$⊢ φ \to (ψ \to (φ \land ψ))$
4	3	aleximi	$⊢ \forall x φ \to (\exists x ψ \to \exists x (φ \land ψ))$
5	2 4	jaoa	$⊢ (\forall x ψ \lor \forall x φ) \to ((\exists x φ \land \exists x ψ) \to \exists x (φ \land ψ))$
6	5	orcoms	$⊢ (\forall x φ \lor \forall x ψ) \to ((\exists x φ \land \exists x ψ) \to \exists x (φ \land ψ))$
7		19.40	$⊢ \exists x (φ \land ψ) \to (\exists x φ \land \exists x ψ)$
8	6 7	impbid1	$⊢ (\forall x φ \lor \forall x ψ) \to ((\exists x φ \land \exists x ψ) \leftrightarrow \exists x (φ \land ψ))$