uunT1

Description: A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 3-Dec-2015) Proof was revised to accommodate a possible future version of df-tru . (Revised by David A. Wheeler, 8-May-2019) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	uunT1.1	⊢ ( ( ⊤ ∧ 𝜑 ) → 𝜓 )
	Assertion	uunT1	⊢ ( 𝜑 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		uunT1.1	⊢ ( ( ⊤ ∧ 𝜑 ) → 𝜓 )
2		orc	⊢ ( 𝜑 → ( 𝜑 ∨ ¬ 𝜑 ) )
3		tru	⊢ ⊤
4		biid	⊢ ( 𝜑 ↔ 𝜑 )
5	3 4	2th	⊢ ( ⊤ ↔ ( 𝜑 ↔ 𝜑 ) )
6		exmid	⊢ ( 𝜑 ∨ ¬ 𝜑 )
7	6	a1i	⊢ ( ( 𝜑 ↔ 𝜑 ) → ( 𝜑 ∨ ¬ 𝜑 ) )
8		biidd	⊢ ( ( 𝜑 ∨ ¬ 𝜑 ) → ( 𝜑 ↔ 𝜑 ) )
9	7 8	impbii	⊢ ( ( 𝜑 ↔ 𝜑 ) ↔ ( 𝜑 ∨ ¬ 𝜑 ) )
10	5 9	bitri	⊢ ( ⊤ ↔ ( 𝜑 ∨ ¬ 𝜑 ) )
11	2 10	sylibr	⊢ ( 𝜑 → ⊤ )
12	11 1	mpancom	⊢ ( 𝜑 → 𝜓 )

Metamath Proof Explorer

Theorem uunT1

Proof