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Metamath Proof Explorer

Theorem daraptiALT

Description: Alternate proof of darapti , shorter but using more axioms. See comment of dariiALT . (Contributed by David A. Wheeler, 27-Aug-2016) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	darapti.maj	⊢ ∀ 𝑥 ( 𝜑 → 𝜓 )
	darapti.min	⊢ ∀ 𝑥 ( 𝜑 → 𝜒 )
	darapti.e	⊢ ∃ 𝑥 𝜑
Assertion	daraptiALT	⊢ ∃ 𝑥 ( 𝜒 ∧ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		darapti.maj	⊢ ∀ 𝑥 ( 𝜑 → 𝜓 )
2		darapti.min	⊢ ∀ 𝑥 ( 𝜑 → 𝜒 )
3		darapti.e	⊢ ∃ 𝑥 𝜑
4	2	spi	⊢ ( 𝜑 → 𝜒 )
5	1	spi	⊢ ( 𝜑 → 𝜓 )
6	4 5	jca	⊢ ( 𝜑 → ( 𝜒 ∧ 𝜓 ) )
7	3 6	eximii	⊢ ∃ 𝑥 ( 𝜒 ∧ 𝜓 )