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	|- ( ( if- ( ch , ph , ps ) <-> ph ) -> ( ta <-> ch ) )
	elimh.2	|- ( ( if- ( ch , ph , ps ) <-> ps ) -> ( ta <-> th ) )
	elimh.3	|- th
Assertion	elimh	|- ta

Proof

Step	Hyp	Ref	Expression
1		elimh.1	|- ( ( if- ( ch , ph , ps ) <-> ph ) -> ( ta <-> ch ) )
2		elimh.2	|- ( ( if- ( ch , ph , ps ) <-> ps ) -> ( ta <-> th ) )
3		elimh.3	|- th
4		ifptru	|- ( ch -> ( if- ( ch , ph , ps ) <-> ph ) )
5	4 1	syl	|- ( ch -> ( ta <-> ch ) )
6	5	ibir	|- ( ch -> ta )
7		ifpfal	|- ( -. ch -> ( if- ( ch , ph , ps ) <-> ps ) )
8	7 2	syl	|- ( -. ch -> ( ta <-> th ) )
9	3 8	mpbiri	|- ( -. ch -> ta )
10	6 9	pm2.61i	|- ta