Theorem pm5.21nd

Description: Eliminate an antecedent implied by each side of a biconditional. Variant of pm5.21ndd . (Contributed by NM, 20-Nov-2005) (Proof shortened by Wolf Lammen, 4-Nov-2013)

	Ref	Expression
Hypotheses	pm5.21nd.1	$$⊢\left(\phi \wedge \psi \right)\to \theta$$
	pm5.21nd.2	$$⊢\left(\phi \wedge \chi \right)\to \theta$$
	pm5.21nd.3	$$⊢\theta \to \left(\psi \leftrightarrow \chi \right)$$
Assertion	pm5.21nd	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		pm5.21nd.1	$$⊢\left(\phi \wedge \psi \right)\to \theta$$
2		pm5.21nd.2	$$⊢\left(\phi \wedge \chi \right)\to \theta$$
3		pm5.21nd.3	$$⊢\theta \to \left(\psi \leftrightarrow \chi \right)$$
4	1	ex	$$⊢\phi \to \left(\psi \to \theta \right)$$
5	2	ex	$$⊢\phi \to \left(\chi \to \theta \right)$$
6	3	a1i	$$⊢\phi \to \left(\theta \to \left(\psi \leftrightarrow \chi \right)\right)$$
7	4 5 6	pm5.21ndd	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$