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Metamath Proof Explorer

Theorem uun2131p1

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	uun2131p1.1	⊢ ( ( ( 𝜑 ∧ 𝜒 ) ∧ ( 𝜑 ∧ 𝜓 ) ) → 𝜃 )
	Assertion	uun2131p1	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		uun2131p1.1	⊢ ( ( ( 𝜑 ∧ 𝜒 ) ∧ ( 𝜑 ∧ 𝜓 ) ) → 𝜃 )
2		ancom	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜑 ∧ 𝜒 ) ) ↔ ( ( 𝜑 ∧ 𝜒 ) ∧ ( 𝜑 ∧ 𝜓 ) ) )
3	2 1	sylbi	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜑 ∧ 𝜒 ) ) → 𝜃 )
4	3	3impdi	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → 𝜃 )