Theorem nfrexg 3309

Description: Bound-variable hypothesis builder for restricted quantification. Usage of this theorem is discouraged because it depends on ax-13 2389. See nfrex 3308 for a version with a disjoint variable condition, but not requiring ax-13 2389. (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
nfrexg.1	⊢ Ⅎ𝑥𝐴
nfrexg.2	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfrexg	⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑

Proof of Theorem nfrexg

Step	Hyp	Ref	Expression
1		nftru 1804	. . 3 ⊢ Ⅎ𝑦⊤
2		nfrexg.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐴)
4		nfrexg.2	. . . 4 ⊢ Ⅎ𝑥𝜑
5	4	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
6	1, 3, 5	nfrexdg 3307	. 2 ⊢ (⊤ → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑)
7	6	mptru 1543	1 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑

Syntax hints:  ⊤wtru 1537  Ⅎwnf 1783  Ⅎwnfc 2960  ∃wrex 3138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-13 2389  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ral 3142  df-rex 3143

This theorem is referenced by:  nfiung  4944