Theorem elimphu

Description: Hypothesis elimination lemma for complex inner product spaces to assist weak deduction theorem. (Contributed by NM, 6-May-2007) (New usage is discouraged.)

	Ref	Expression
Assertion	elimphu	$$⊢if\left(U\in {CPreHil}_{\mathrm{OLD}},U,⟨⟨+,\times ⟩,\mathrm{abs}⟩\right)\in {CPreHil}_{\mathrm{OLD}}$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢⟨⟨+,\times ⟩,\mathrm{abs}⟩=⟨⟨+,\times ⟩,\mathrm{abs}⟩$$
2	1	cncph	$$⊢⟨⟨+,\times ⟩,\mathrm{abs}⟩\in {CPreHil}_{\mathrm{OLD}}$$
3	2	elimel	$$⊢if\left(U\in {CPreHil}_{\mathrm{OLD}},U,⟨⟨+,\times ⟩,\mathrm{abs}⟩\right)\in {CPreHil}_{\mathrm{OLD}}$$