Theorem orim12d

Description: Disjoin antecedents and consequents in a deduction. See orim12dALT for a proof which does not depend on df-an . (Contributed by NM, 10-May-1994)

	Ref	Expression
Hypotheses	orim12d.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
	orim12d.2	⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )
Assertion	orim12d	⊢ ( 𝜑 → ( ( 𝜓 ∨ 𝜃 ) → ( 𝜒 ∨ 𝜏 ) ) )

Proof

Step	Hyp	Ref	Expression
1		orim12d.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		orim12d.2	⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )
3		pm3.48	⊢ ( ( ( 𝜓 → 𝜒 ) ∧ ( 𝜃 → 𝜏 ) ) → ( ( 𝜓 ∨ 𝜃 ) → ( 𝜒 ∨ 𝜏 ) ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( ( 𝜓 ∨ 𝜃 ) → ( 𝜒 ∨ 𝜏 ) ) )