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Metamath Proof Explorer

Theorem in2an

Description: The virtual deduction introduction rule converting the second conjunct of the second virtual hypothesis into the antecedent of the conclusion. expd is the non-virtual deduction form of in2an . (Contributed by Alan Sare, 30-Jun-2012) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	in2an.1	⊢ ( 𝜑 , ( 𝜓 ∧ 𝜒 ) ▶ 𝜃 )
	Assertion	in2an	⊢ ( 𝜑 , 𝜓 ▶ ( 𝜒 → 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		in2an.1	⊢ ( 𝜑 , ( 𝜓 ∧ 𝜒 ) ▶ 𝜃 )
2	1	dfvd2i	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜒 ) → 𝜃 ) )
3	2	expd	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜃 ) ) )
4	3	dfvd2ir	⊢ ( 𝜑 , 𝜓 ▶ ( 𝜒 → 𝜃 ) )