Theorem elimh

Description: Hypothesis builder for 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	elimh.1	$$⊢\left(if-\left(\chi ,\phi ,\psi \right)\leftrightarrow \phi \right)\to \left(\tau \leftrightarrow \chi \right)$$
	elimh.2	$$⊢\left(if-\left(\chi ,\phi ,\psi \right)\leftrightarrow \psi \right)\to \left(\tau \leftrightarrow \theta \right)$$
	elimh.3	$$⊢\theta$$
Assertion	elimh	$$⊢\tau$$

Proof

Step	Hyp	Ref	Expression
1		elimh.1	$$⊢\left(if-\left(\chi ,\phi ,\psi \right)\leftrightarrow \phi \right)\to \left(\tau \leftrightarrow \chi \right)$$
2		elimh.2	$$⊢\left(if-\left(\chi ,\phi ,\psi \right)\leftrightarrow \psi \right)\to \left(\tau \leftrightarrow \theta \right)$$
3		elimh.3	$$⊢\theta$$
4		ifptru	$$⊢\chi \to \left(if-\left(\chi ,\phi ,\psi \right)\leftrightarrow \phi \right)$$
5	4 1	syl	$$⊢\chi \to \left(\tau \leftrightarrow \chi \right)$$
6	5	ibir	$$⊢\chi \to \tau$$
7		ifpfal	$$⊢\neg \chi \to \left(if-\left(\chi ,\phi ,\psi \right)\leftrightarrow \psi \right)$$
8	7 2	syl	$$⊢\neg \chi \to \left(\tau \leftrightarrow \theta \right)$$
9	3 8	mpbiri	$$⊢\neg \chi \to \tau$$
10	6 9	pm2.61i	$$⊢\tau$$