Theorem elimdhyp

Description: Version of elimhyp where the hypothesis is deduced from the final antecedent. See divalg for an example of its use. (Contributed by Paul Chapman, 25-Mar-2008)

	Ref	Expression
Hypotheses	elimdhyp.1	$$⊢\phi \to \psi$$
	elimdhyp.2	$$⊢A=if\left(\phi ,A,B\right)\to \left(\psi \leftrightarrow \chi \right)$$
	elimdhyp.3	$$⊢B=if\left(\phi ,A,B\right)\to \left(\theta \leftrightarrow \chi \right)$$
	elimdhyp.4	$$⊢\theta$$
Assertion	elimdhyp	$$⊢\chi$$

Proof

Step	Hyp	Ref	Expression
1		elimdhyp.1	$$⊢\phi \to \psi$$
2		elimdhyp.2	$$⊢A=if\left(\phi ,A,B\right)\to \left(\psi \leftrightarrow \chi \right)$$
3		elimdhyp.3	$$⊢B=if\left(\phi ,A,B\right)\to \left(\theta \leftrightarrow \chi \right)$$
4		elimdhyp.4	$$⊢\theta$$
5		iftrue	$$⊢\phi \to if\left(\phi ,A,B\right)=A$$
6	5	eqcomd	$$⊢\phi \to A=if\left(\phi ,A,B\right)$$
7	6 2	syl	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
8	1 7	mpbid	$$⊢\phi \to \chi$$
9		iffalse	$$⊢\neg \phi \to if\left(\phi ,A,B\right)=B$$
10	9	eqcomd	$$⊢\neg \phi \to B=if\left(\phi ,A,B\right)$$
11	10 3	syl	$$⊢\neg \phi \to \left(\theta \leftrightarrow \chi \right)$$
12	4 11	mpbii	$$⊢\neg \phi \to \chi$$
13	8 12	pm2.61i	$$⊢\chi$$