Theorem ex-natded5.3-2 30428

Description: A more efficient proof of Theorem 5.3 of [Clemente] p. 16. Compare with ex-natded5.3 30427 and ex-natded5.3i 30429. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ex-natded5.3.1	⊢ (𝜑 → (𝜓 → 𝜒))
ex-natded5.3.2	⊢ (𝜑 → (𝜒 → 𝜃))

Assertion

Ref	Expression
ex-natded5.3-2	⊢ (𝜑 → (𝜓 → (𝜒 ∧ 𝜃)))

Proof of Theorem ex-natded5.3-2

Step	Hyp	Ref	Expression
1		ex-natded5.3.1	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
2		ex-natded5.3.2	. . 3 ⊢ (𝜑 → (𝜒 → 𝜃))
3	1, 2	syld 47	. 2 ⊢ (𝜑 → (𝜓 → 𝜃))
4	1, 3	jcad 512	1 ⊢ (𝜑 → (𝜓 → (𝜒 ∧ 𝜃)))

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)