Theorem a2and

Description: Deduction distributing a conjunction as embedded antecedent. (Contributed by AV, 25-Oct-2019) (Proof shortened by Wolf Lammen, 19-Jan-2020)

	Ref	Expression
Hypotheses	a2and.1	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜌 ) → ( 𝜏 → 𝜃 ) ) )
	a2and.2	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜌 ) → 𝜒 ) )
Assertion	a2and	⊢ ( 𝜑 → ( ( ( 𝜓 ∧ 𝜒 ) → 𝜏 ) → ( ( 𝜓 ∧ 𝜌 ) → 𝜃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		a2and.1	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜌 ) → ( 𝜏 → 𝜃 ) ) )
2		a2and.2	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜌 ) → 𝜒 ) )
3	2	expd	⊢ ( 𝜑 → ( 𝜓 → ( 𝜌 → 𝜒 ) ) )
4	3	imdistand	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜌 ) → ( 𝜓 ∧ 𝜒 ) ) )
5		imim1	⊢ ( ( ( 𝜓 ∧ 𝜒 ) → 𝜏 ) → ( ( 𝜏 → 𝜃 ) → ( ( 𝜓 ∧ 𝜒 ) → 𝜃 ) ) )
6	5	com3l	⊢ ( ( 𝜏 → 𝜃 ) → ( ( 𝜓 ∧ 𝜒 ) → ( ( ( 𝜓 ∧ 𝜒 ) → 𝜏 ) → 𝜃 ) ) )
7	1 4 6	syl6c	⊢ ( 𝜑 → ( ( 𝜓 ∧ 𝜌 ) → ( ( ( 𝜓 ∧ 𝜒 ) → 𝜏 ) → 𝜃 ) ) )
8	7	com23	⊢ ( 𝜑 → ( ( ( 𝜓 ∧ 𝜒 ) → 𝜏 ) → ( ( 𝜓 ∧ 𝜌 ) → 𝜃 ) ) )