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Metamath Proof Explorer

Theorem spimew

Description: Existential introduction, using implicit substitution. Compare Lemma 14 of Tarski p. 70. (Contributed by NM, 7-Aug-1994) (Proof shortened by Wolf Lammen, 22-Oct-2023)

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

Proof

Step	Hyp	Ref	Expression
1		spimew.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2		spimew.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
3		ax6v	⊢ ¬ ∀ 𝑥 ¬ 𝑥 = 𝑦
4	2	speimfw	⊢ ( ¬ ∀ 𝑥 ¬ 𝑥 = 𝑦 → ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )
5	3 1 4	mpsyl	⊢ ( 𝜑 → ∃ 𝑥 𝜓 )