Theorem re1tbw2

Description: tbw-ax2 rederived from merco2 . (Contributed by Anthony Hart, 16-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	re1tbw2	$$⊢\phi \to \left(\psi \to \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		mercolem1	$$⊢\left(\left(\phi \to \phi \right)\to \phi \right)\to \left(\phi \to \left(\psi \to \phi \right)\right)$$
2		mercolem1	$$⊢\left(\left(\left(\phi \to \phi \right)\to \phi \right)\to \left(\phi \to \left(\psi \to \phi \right)\right)\right)\to \left(\phi \to \left(\psi \to \left(\phi \to \left(\psi \to \phi \right)\right)\right)\right)$$
3	1 2	ax-mp	$$⊢\phi \to \left(\psi \to \left(\phi \to \left(\psi \to \phi \right)\right)\right)$$
4		mercolem6	$$⊢\left(\phi \to \left(\psi \to \left(\phi \to \left(\psi \to \phi \right)\right)\right)\right)\to \left(\psi \to \left(\phi \to \left(\psi \to \phi \right)\right)\right)$$
5	3 4	ax-mp	$$⊢\psi \to \left(\phi \to \left(\psi \to \phi \right)\right)$$
6		mercolem6	$$⊢\left(\psi \to \left(\phi \to \left(\psi \to \phi \right)\right)\right)\to \left(\phi \to \left(\psi \to \phi \right)\right)$$
7	5 6	ax-mp	$$⊢\phi \to \left(\psi \to \phi \right)$$