Theorem nfralxy 2471

Description: Not-free for restricted universal quantification where 𝑥 and 𝑦 are distinct. See nfralya 2473 for a version with 𝑦 and 𝐴 distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)

Hypotheses

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

Assertion

Ref	Expression
nfralxy	⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfralxy

Step	Hyp	Ref	Expression
1		nftru 1442	. . 3 ⊢ Ⅎ𝑦⊤
2		nfralxy.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	a1i 9	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐴)
4		nfralxy.2	. . . 4 ⊢ Ⅎ𝑥𝜑
5	4	a1i 9	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
6	1, 3, 5	nfraldxy 2467	. 2 ⊢ (⊤ → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑)
7	6	mptru 1340	1 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑

Syntax hints:  ⊤wtru 1332  Ⅎwnf 1436  Ⅎwnfc 2268  ∀wral 2416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421

This theorem is referenced by:  nfra2xy  2475  rspc2  2800  sbcralt  2985  sbcralg  2987  raaanlem  3468  nfint  3781  nfiinxy  3840  nfpo  4223  nfso  4224  nfse  4263  nffrfor  4270  nfwe  4277  ralxpf  4685  funimaexglem  5206  fun11iun  5388  dff13f  5671  nfiso  5707  mpoeq123  5830  nfofr  5988  fmpox  6098  nfrecs  6204  xpf1o  6738  ac6sfi  6792  ismkvnex  7029  lble  8705  fzrevral  9885  nfsum1  11125  nfsum  11126  fsum2dlemstep  11203  fisumcom2  11207  nfcprod1  11323  nfcprod  11324  bezoutlemmain  11686  cnmpt21  12460  setindis  13165  bdsetindis  13167  isomninnlem  13225