Theorem nfral 3363

Description: Bound-variable hypothesis builder for restricted quantification. Usage of this theorem is discouraged because it depends on ax-13 2404. Use the weaker nfralw 3312 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 1834	. . 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 3361	. 2 ⊢ (⊤ → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑)
7	6	mptru 1577	1 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑

Syntax hints:  ⊤wtru 1571  Ⅎwnf 1813  Ⅎwnfc 2910  ∀wral 3079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-13 2404  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080

This theorem is referenced by:  nfra2  3365  nfiing  4991  opreu2reuALT  32823  eliuniincex  45827  cbvral2  47840