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Metamath Proof Explorer

Theorem spimedv

Description: Deduction version of spimev . Version of spimed with a disjoint variable condition, which does not require ax-13 . See spime for a non-deduction version. (Contributed by NM, 14-May-1993) (Revised by BJ, 31-May-2019)

	Ref	Expression
Hypotheses	spimedv.1	⊢ ( 𝜒 → Ⅎ 𝑥 𝜑 )
	spimedv.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
Assertion	spimedv	⊢ ( 𝜒 → ( 𝜑 → ∃ 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		spimedv.1	⊢ ( 𝜒 → Ⅎ 𝑥 𝜑 )
2		spimedv.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
3	1	nf5rd	⊢ ( 𝜒 → ( 𝜑 → ∀ 𝑥 𝜑 ) )
4		ax6ev	⊢ ∃ 𝑥 𝑥 = 𝑦
5	4 2	eximii	⊢ ∃ 𝑥 ( 𝜑 → 𝜓 )
6	5	19.35i	⊢ ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 )
7	3 6	syl6	⊢ ( 𝜒 → ( 𝜑 → ∃ 𝑥 𝜓 ) )