Theorem a2and 548

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

Hypotheses

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

Assertion

Ref	Expression
a2and	⊢ (𝜑 → (((𝜓 ∧ 𝜒) → 𝜏) → ((𝜓 ∧ 𝜌) → 𝜃)))

Proof of Theorem a2and

Step	Hyp	Ref	Expression
1		a2and.2	. . . . . . 7 ⊢ (𝜑 → ((𝜓 ∧ 𝜌) → 𝜒))
2	1	expd 256	. . . . . 6 ⊢ (𝜑 → (𝜓 → (𝜌 → 𝜒)))
3	2	imdistand 444	. . . . 5 ⊢ (𝜑 → ((𝜓 ∧ 𝜌) → (𝜓 ∧ 𝜒)))
4	3	imp 123	. . . 4 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜌)) → (𝜓 ∧ 𝜒))
5		a2and.1	. . . . 5 ⊢ (𝜑 → ((𝜓 ∧ 𝜌) → (𝜏 → 𝜃)))
6	5	imp 123	. . . 4 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜌)) → (𝜏 → 𝜃))
7	4, 6	embantd 56	. . 3 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜌)) → (((𝜓 ∧ 𝜒) → 𝜏) → 𝜃))
8	7	ex 114	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜌) → (((𝜓 ∧ 𝜒) → 𝜏) → 𝜃)))
9	8	com23 78	1 ⊢ (𝜑 → (((𝜓 ∧ 𝜒) → 𝜏) → ((𝜓 ∧ 𝜌) → 𝜃)))

Syntax hints:   → wi 4   ∧ wa 103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)