Theorem dedt

Description: The weak deduction theorem. For more information, see the Weak Deduction Theorem page mmdeduction.html . (Contributed by NM, 26-Jun-2002) Revised to use the conditional operator. (Revised by BJ, 30-Sep-2019) Commute consequent. (Revised by Steven Nguyen, 27-Apr-2023)

	Ref	Expression
Hypotheses	dedt.1	$$⊢\left(if-\left(\chi ,\phi ,\psi \right)\leftrightarrow \phi \right)\to \left(\tau \leftrightarrow \theta \right)$$
	dedt.2	$$⊢\tau$$
Assertion	dedt	$$⊢\chi \to \theta$$

Proof

Step	Hyp	Ref	Expression
1		dedt.1	$$⊢\left(if-\left(\chi ,\phi ,\psi \right)\leftrightarrow \phi \right)\to \left(\tau \leftrightarrow \theta \right)$$
2		dedt.2	$$⊢\tau$$
3		ifptru	$$⊢\chi \to \left(if-\left(\chi ,\phi ,\psi \right)\leftrightarrow \phi \right)$$
4	2 1	mpbii	$$⊢\left(if-\left(\chi ,\phi ,\psi \right)\leftrightarrow \phi \right)\to \theta$$
5	3 4	syl	$$⊢\chi \to \theta$$