Theorem ex-natded5.2-2 28178

Description: A more efficient proof of Theorem 5.2 of [Clemente] p. 15. Compare with ex-natded5.2 28177 and ex-natded5.2i 28179. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ex-natded5.2.1	⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))
ex-natded5.2.2	⊢ (𝜑 → (𝜒 → 𝜓))
ex-natded5.2.3	⊢ (𝜑 → 𝜒)

Assertion

Ref	Expression
ex-natded5.2-2	⊢ (𝜑 → 𝜃)

Proof of Theorem ex-natded5.2-2

Step	Hyp	Ref	Expression
1		ex-natded5.2.3	. . 3 ⊢ (𝜑 → 𝜒)
2		ex-natded5.2.2	. . 3 ⊢ (𝜑 → (𝜒 → 𝜓))
3	1, 2	mpd 15	. 2 ⊢ (𝜑 → 𝜓)
4		ex-natded5.2.1	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))
5	3, 1, 4	mp2and 697	1 ⊢ (𝜑 → 𝜃)

Syntax hints:   → wi 4   ∧ wa 398

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 209  df-an 399

This theorem is referenced by: (None)