Theorem adh-minim-idALT

Description: Derivation of id (reflexivity of implication, PM *2.08 WhiteheadRussell p. 101) from adh-minim-ax1 , adh-minim-ax2 , and ax-mp . It uses the derivation written DD211 in D-notation. (See head comment for an explanation.) Polish prefix notation: Cpp . (Contributed by ADH, 10-Nov-2023) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	adh-minim-idALT	$$⊢\phi \to \phi$$

Proof

Step	Hyp	Ref	Expression
1		adh-minim-ax1	$$⊢\phi \to \left(\psi \to \phi \right)$$
2		adh-minim-ax1	$$⊢\phi \to \left(\left(\psi \to \phi \right)\to \phi \right)$$
3		adh-minim-ax2	$$⊢\left(\phi \to \left(\left(\psi \to \phi \right)\to \phi \right)\right)\to \left(\left(\phi \to \left(\psi \to \phi \right)\right)\to \left(\phi \to \phi \right)\right)$$
4	2 3	ax-mp	$$⊢\left(\phi \to \left(\psi \to \phi \right)\right)\to \left(\phi \to \phi \right)$$
5	1 4	ax-mp	$$⊢\phi \to \phi$$