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Metamath Proof Explorer

Theorem spimed

Description: Deduction version of spime . (Contributed by NM, 14-May-1993) (Revised by Mario Carneiro, 3-Oct-2016) (Proof shortened by Wolf Lammen, 19-Feb-2018) Usage of this theorem is discouraged because it depends on ax-13 . Use spimedv instead. (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		spimed.1	⊢ ( 𝜒 → Ⅎ 𝑥 𝜑 )
2		spimed.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
3	1	nf5rd	⊢ ( 𝜒 → ( 𝜑 → ∀ 𝑥 𝜑 ) )
4		ax6e	⊢ ∃ 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 ) )