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Metamath Proof Explorer

Theorem ex-natded5.7-2

Description: A more efficient proof of Theorem 5.7 of Clemente p. 19. Compare with ex-natded5.7 . (Contributed by Mario Carneiro, 9-Feb-2017) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	ex-natded5.7.1	⊢ ( 𝜑 → ( 𝜓 ∨ ( 𝜒 ∧ 𝜃 ) ) )
	Assertion	ex-natded5.7-2	⊢ ( 𝜑 → ( 𝜓 ∨ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		ex-natded5.7.1	⊢ ( 𝜑 → ( 𝜓 ∨ ( 𝜒 ∧ 𝜃 ) ) )
2		simpl	⊢ ( ( 𝜒 ∧ 𝜃 ) → 𝜒 )
3	2	orim2i	⊢ ( ( 𝜓 ∨ ( 𝜒 ∧ 𝜃 ) ) → ( 𝜓 ∨ 𝜒 ) )
4	1 3	syl	⊢ ( 𝜑 → ( 𝜓 ∨ 𝜒 ) )