Theorem nfral 3153

Description: Bound-variable hypothesis builder for restricted quantification. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker nfralw 3151 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 1807	. . 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 3150	. 2 ⊢ (⊤ → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑)
7	6	mptru 1546	1 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑

Syntax hints:  ⊤wtru 1540  Ⅎwnf 1786  Ⅎwnfc 2887  ∀wral 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2372  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069

This theorem is referenced by:  nfra2  3157  nfiing  4957  opreu2reuALT  30825  eliuniincex  42659  cbvral2  44595