Theorem orim12da

Description: Deduce a disjunction from another one. Variation on orim12d . (Contributed by Thierry Arnoux, 18-May-2025)

	Ref	Expression
Hypotheses	orim12da.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜃 )
	orim12da.2	⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜏 )
	orim12da.3	⊢ ( 𝜑 → ( 𝜓 ∨ 𝜒 ) )
Assertion	orim12da	⊢ ( 𝜑 → ( 𝜃 ∨ 𝜏 ) )

Proof

Step	Hyp	Ref	Expression
1		orim12da.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜃 )
2		orim12da.2	⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜏 )
3		orim12da.3	⊢ ( 𝜑 → ( 𝜓 ∨ 𝜒 ) )
4	1	ex	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )
5	2	ex	⊢ ( 𝜑 → ( 𝜒 → 𝜏 ) )
6	4 5	orim12d	⊢ ( 𝜑 → ( ( 𝜓 ∨ 𝜒 ) → ( 𝜃 ∨ 𝜏 ) ) )
7	3 6	mpd	⊢ ( 𝜑 → ( 𝜃 ∨ 𝜏 ) )