Theorem nfrex 3360

Description: Bound-variable hypothesis builder for restricted quantification. Usage of this theorem is discouraged because it depends on ax-13 2401. See nfrexw 3310 for a version with a disjoint variable condition, but not requiring ax-13 2401. (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.)

Hypotheses

Ref	Expression
nfral.1	⊢ Ⅎ𝑥𝐴
nfral.2	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfrex	⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑

Proof of Theorem nfrex

Step	Hyp	Ref	Expression
1		nftru 1836	. . 3 ⊢ Ⅎ𝑦⊤
2		nfral.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐴)
4		nfral.2	. . . 4 ⊢ Ⅎ𝑥𝜑
5	4	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
6	1, 3, 5	nfrexd 3358	. 2 ⊢ (⊤ → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑)
7	6	mptru 1576	1 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑

This proof depends on syntax axioms:  ⊤wtru 1570  Ⅎwnf 1815  Ⅎwnfc 2907  ∃wrex 3086

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-13 2401  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087

This theorem is used by:  nfiung  4983