Theorem nfral 3338

Description: Bound-variable hypothesis builder for restricted quantification. Usage of this theorem is discouraged because it depends on ax-13 2380. Use the weaker nfralw 3286 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 1811	. . 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 3336	. 2 ⊢ (⊤ → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑)
7	6	mptru 1554	1 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑

Syntax hints:  ⊤wtru 1548  Ⅎwnf 1790  Ⅎwnfc 2886  ∀wral 3053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-13 2380  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ral 3054

This theorem is referenced by:  nfra2  3340  nfiing  4956  opreu2reuALT  32564  eliuniincex  45556  cbvral2  47566