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Metamath Proof Explorer

Theorem 19.8a

Description: If a wff is true, it is true for at least one instance. Special case of Theorem 19.8 of Margaris p. 89. See 19.8v for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 9-Jan-1993) Allow a shortening of sp . (Revised by Wolf Lammen, 13-Jan-2018) (Proof shortened by Wolf Lammen, 8-Dec-2019)

		Ref	Expression
	Assertion	19.8a	⊢ ( 𝜑 → ∃ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		ax12v	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
2		alequexv	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ∃ 𝑥 𝜑 )
3	1 2	syl6	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → ∃ 𝑥 𝜑 ) )
4		ax6evr	⊢ ∃ 𝑦 𝑥 = 𝑦
5	3 4	exlimiiv	⊢ ( 𝜑 → ∃ 𝑥 𝜑 )