This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem 3impdirp1

Description: A deduction unionizing a non-unionized collection of virtual hypotheses. Commuted version of 3impdir . (Contributed by Alan Sare, 4-Feb-2017) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	3impdirp1.1	⊢ ( ( ( 𝜒 ∧ 𝜓 ) ∧ ( 𝜑 ∧ 𝜓 ) ) → 𝜃 )
	Assertion	3impdirp1	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜓 ) → 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		3impdirp1.1	⊢ ( ( ( 𝜒 ∧ 𝜓 ) ∧ ( 𝜑 ∧ 𝜓 ) ) → 𝜃 )
2		ancom	⊢ ( ( ( 𝜒 ∧ 𝜓 ) ∧ ( 𝜑 ∧ 𝜓 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜒 ∧ 𝜓 ) ) )
3	2 1	sylbir	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜒 ∧ 𝜓 ) ) → 𝜃 )
4	3	3impdir	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜓 ) → 𝜃 )