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Metamath Proof Explorer

Theorem nfrex

Description: Bound-variable hypothesis builder for restricted quantification. Usage of this theorem is discouraged because it depends on ax-13 . See nfrexw for a version with a disjoint variable condition, but not requiring ax-13 . (Contributed by NM, 1-Sep-1999) (Revised by Mario Carneiro, 7-Oct-2016) (Proof shortened by Wolf Lammen, 30-Dec-2019) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nfral.1	⊢ Ⅎ 𝑥 𝐴
	nfral.2	⊢ Ⅎ 𝑥 𝜑
Assertion	nfrex	⊢ Ⅎ 𝑥 ∃ 𝑦 ∈ 𝐴 𝜑

Proof

Step	Hyp	Ref	Expression
1		nfral.1	⊢ Ⅎ 𝑥 𝐴
2		nfral.2	⊢ Ⅎ 𝑥 𝜑
3		nftru	⊢ Ⅎ 𝑦 ⊤
4	1	a1i	⊢ ( ⊤ → Ⅎ 𝑥 𝐴 )
5	2	a1i	⊢ ( ⊤ → Ⅎ 𝑥 𝜑 )
6	3 4 5	nfrexd	⊢ ( ⊤ → Ⅎ 𝑥 ∃ 𝑦 ∈ 𝐴 𝜑 )
7	6	mptru	⊢ Ⅎ 𝑥 ∃ 𝑦 ∈ 𝐴 𝜑