Theorem nfral 2667

Description: Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.)

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 1554	. . 3 ⊢ Ⅎy ⊤
2		nfral.1	. . . 4 ⊢ ℲxA
3	2	a1i 10	. . 3 ⊢ ( ⊤ → ℲxA)
4		nfral.2	. . . 4 ⊢ Ⅎxφ
5	4	a1i 10	. . 3 ⊢ ( ⊤ → Ⅎxφ)
6	1, 3, 5	nfrald 2665	. 2 ⊢ ( ⊤ → Ⅎx∀y ∈ A φ)
7	6	trud 1323	1 ⊢ Ⅎx∀y ∈ A φ

Syntax hints:   ⊤ wtru 1316  Ⅎwnf 1544  Ⅎwnfc 2476  ∀wral 2614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619

This theorem is referenced by:  nfra2  2668  nfrex  2669  rspc2  2960  sbcralt  3118  sbcralg  3120  raaan  3657  nfint  3936  nfiin  3996  ralxpf  4827  fun11iun  5305  dff13f  5472  nfiso  5487  mpt2eq123  5661  fmpt2x  5730