Theorem e12an

Description: Conjunction form of e12 (see syl6an ). (Contributed by Alan Sare, 11-Jun-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	e12an.1	$$⊢\left(\phi \to \psi \right)$$
	e12an.2	$$⊢\left(\phi ,\chi \to \theta \right)$$
	e12an.3	$$⊢\left(\psi \wedge \theta \right)\to \tau$$
Assertion	e12an	$$⊢\left(\phi ,\chi \to \tau \right)$$

Proof

Step	Hyp	Ref	Expression
1		e12an.1	$$⊢\left(\phi \to \psi \right)$$
2		e12an.2	$$⊢\left(\phi ,\chi \to \theta \right)$$
3		e12an.3	$$⊢\left(\psi \wedge \theta \right)\to \tau$$
4	3	ex	$$⊢\psi \to \left(\theta \to \tau \right)$$
5	1 2 4	e12	$$⊢\left(\phi ,\chi \to \tau \right)$$