Theorem keephyp

Description: Transform a hypothesis ps that we want to keep (but contains the same class variable A used in the eliminated hypothesis) for use with the weak deduction theorem. (Contributed by NM, 15-May-1999)

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

Proof

Step	Hyp	Ref	Expression
1		keephyp.1	$$⊢A=if\left(\phi ,A,B\right)\to \left(\psi \leftrightarrow \theta \right)$$
2		keephyp.2	$$⊢B=if\left(\phi ,A,B\right)\to \left(\chi \leftrightarrow \theta \right)$$
3		keephyp.3	$$⊢\psi$$
4		keephyp.4	$$⊢\chi$$
5	1 2	ifboth	$$⊢\left(\psi \wedge \chi \right)\to \theta$$
6	3 4 5	mp2an	$$⊢\theta$$