Theorem nfral 3361

Description: Bound-variable hypothesis builder for restricted quantification. Usage of this theorem is discouraged because it depends on ax-13 2403. Use the weaker nfralw 3309 when possible. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
nfral	⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑

Proof of Theorem nfral

Step	Hyp	Ref	Expression
1		nftru 1824	. . 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	nfrald 3359	. 2 ⊢ (⊤ → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑)
7	6	mptru 1567	1 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑

Syntax hints:  ⊤wtru 1561  Ⅎwnf 1803  Ⅎwnfc 2909  ∀wral 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-13 2403  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-nf 1804  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3077

This theorem is referenced by:  nfra2  3363  nfiing  4984  opreu2reuALT  32673  eliuniincex  45684  cbvral2  47694