Metamath Proof Explorer

Theorem uunT12p3

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	uunT12p3.1	$⊢ (ψ \land ⊤ \land φ) \to χ$
	Assertion	uunT12p3	$⊢ (φ \land ψ) \to χ$

Proof

Step	Hyp	Ref	Expression
1		uunT12p3.1	$⊢ (ψ \land ⊤ \land φ) \to χ$
2		3ancoma	$⊢ (ψ \land ⊤ \land φ) \leftrightarrow (⊤ \land ψ \land φ)$
3		3anass	$⊢ (⊤ \land ψ \land φ) \leftrightarrow (⊤ \land (ψ \land φ))$
4	2 3	bitri	$⊢ (ψ \land ⊤ \land φ) \leftrightarrow (⊤ \land (ψ \land φ))$
5		truan	$⊢ (⊤ \land (ψ \land φ)) \leftrightarrow (ψ \land φ)$
6	4 5	bitri	$⊢ (ψ \land ⊤ \land φ) \leftrightarrow (ψ \land φ)$
7		ancom	$⊢ (φ \land ψ) \leftrightarrow (ψ \land φ)$
8	6 7	bitr4i	$⊢ (ψ \land ⊤ \land φ) \leftrightarrow (φ \land ψ)$
9	8 1	sylbir	$⊢ (φ \land ψ) \to χ$