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Metamath Proof Explorer

Theorem spime

Description: Existential introduction, using implicit substitution. Compare Lemma 14 of Tarski p. 70. See spimew and spimevw for weaker versions requiring fewer axioms. (Contributed by NM, 7-Aug-1994) (Revised by Mario Carneiro, 3-Oct-2016) (Proof shortened by Wolf Lammen, 6-Mar-2018) Usage of this theorem is discouraged because it depends on ax-13 . Use spimefv instead. (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		spime.1	⊢ Ⅎ 𝑥 𝜑
2		spime.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
3	1	a1i	⊢ ( ⊤ → Ⅎ 𝑥 𝜑 )
4	3 2	spimed	⊢ ( ⊤ → ( 𝜑 → ∃ 𝑥 𝜓 ) )
5	4	mptru	⊢ ( 𝜑 → ∃ 𝑥 𝜓 )