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Metamath Proof Explorer

Theorem ex-natded5.3-2

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

	Ref	Expression
Hypotheses	ex-natded5.3.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
	ex-natded5.3.2	⊢ ( 𝜑 → ( 𝜒 → 𝜃 ) )
Assertion	ex-natded5.3-2	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 ∧ 𝜃 ) ) )

Proof

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