Metamath Proof Explorer

Theorem uun2221p1

Description: A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	uun2221p1.1	$⊢ (φ \land (ψ \land φ) \land φ) \to χ$
	Assertion	uun2221p1	$⊢ (ψ \land φ) \to χ$

Proof

Step	Hyp	Ref	Expression
1		uun2221p1.1	$⊢ (φ \land (ψ \land φ) \land φ) \to χ$
2		3anrot	$⊢ (φ \land φ \land (ψ \land φ)) \leftrightarrow (φ \land (ψ \land φ) \land φ)$
3	2	imbi1i	$⊢ ((φ \land φ \land (ψ \land φ)) \to χ) \leftrightarrow ((φ \land (ψ \land φ) \land φ) \to χ)$
4	1 3	mpbir	$⊢ (φ \land φ \land (ψ \land φ)) \to χ$
5		3anass	$⊢ (φ \land φ \land (ψ \land φ)) \leftrightarrow (φ \land (φ \land (ψ \land φ)))$
6		anabs5	$⊢ (φ \land (φ \land (ψ \land φ))) \leftrightarrow (φ \land (ψ \land φ))$
7	5 6	bitri	$⊢ (φ \land φ \land (ψ \land φ)) \leftrightarrow (φ \land (ψ \land φ))$
8		ancom	$⊢ (φ \land ψ) \leftrightarrow (ψ \land φ)$
9	8	anbi2i	$⊢ (φ \land (φ \land ψ)) \leftrightarrow (φ \land (ψ \land φ))$
10	7 9	bitr4i	$⊢ (φ \land φ \land (ψ \land φ)) \leftrightarrow (φ \land (φ \land ψ))$
11		anabs5	$⊢ (φ \land (φ \land ψ)) \leftrightarrow (φ \land ψ)$
12	11 8	bitri	$⊢ (φ \land (φ \land ψ)) \leftrightarrow (ψ \land φ)$
13	10 12	bitri	$⊢ (φ \land φ \land (ψ \land φ)) \leftrightarrow (ψ \land φ)$
14	13	imbi1i	$⊢ ((φ \land φ \land (ψ \land φ)) \to χ) \leftrightarrow ((ψ \land φ) \to χ)$
15	4 14	mpbi	$⊢ (ψ \land φ) \to χ$