Theorem jca3 38857

Description: Inference conjoining the consequents of two implications. (Contributed by Rodolfo Medina, 14-Oct-2010.)

Hypotheses

Ref	Expression
jca3.1	⊢ (𝜑 → (𝜓 → 𝜒))
jca3.2	⊢ (𝜃 → 𝜏)

Assertion

Ref	Expression
jca3	⊢ (𝜑 → (𝜓 → (𝜃 → (𝜒 ∧ 𝜏))))

Proof of Theorem jca3

Step	Hyp	Ref	Expression
1		jca3.1	. . . . 5 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	imp 406	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
3	2	a1d 25	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜃 → 𝜒))
4		jca3.2	. . 3 ⊢ (𝜃 → 𝜏)
5	3, 4	jca2 513	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜃 → (𝜒 ∧ 𝜏)))
6	5	ex 412	1 ⊢ (𝜑 → (𝜓 → (𝜃 → (𝜒 ∧ 𝜏))))

Syntax hints:   → wi 4   ∧ wa 395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-an 396

This theorem is referenced by: (None)