Theorem elimph

Description: Hypothesis elimination lemma for complex inner product spaces to assist weak deduction theorem. (Contributed by NM, 27-Apr-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	elimph.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
	elimph.5	$$⊢Z=0_{\mathrm{vec}}\left(U\right)$$
	elimph.6	$$⊢U\in {CPreHil}_{\mathrm{OLD}}$$
Assertion	elimph	$$⊢if\left(A\in X,A,Z\right)\in X$$

Proof

Step	Hyp	Ref	Expression
1		elimph.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
2		elimph.5	$$⊢Z=0_{\mathrm{vec}}\left(U\right)$$
3		elimph.6	$$⊢U\in {CPreHil}_{\mathrm{OLD}}$$
4	3	phnvi	$$⊢U\in \mathrm{NrmCVec}$$
5	1 2 4	elimnv	$$⊢if\left(A\in X,A,Z\right)\in X$$