Theorem nfralxy 2425

Description: Not-free for restricted universal quantification where 𝑥 and 𝑦 are distinct. See nfralya 2427 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 1407	. . 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 2421	. 2 ⊢ (⊤ → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑)
7	6	mptru 1305	1 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑

Syntax hints:  ⊤wtru 1297  Ⅎwnf 1401  Ⅎwnfc 2222  ∀wral 2370

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-4 1452  ax-17 1471  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375

This theorem is referenced by:  nfra2xy  2429  rspc2  2746  sbcralt  2929  sbcralg  2931  raaanlem  3407  nfint  3720  nfiinxy  3779  nfpo  4152  nfso  4153  nfse  4192  nffrfor  4199  nfwe  4206  ralxpf  4613  funimaexglem  5131  fun11iun  5309  dff13f  5587  nfiso  5623  mpt2eq123  5746  nfofr  5900  fmpt2x  6008  nfrecs  6110  xpf1o  6640  ac6sfi  6694  lble  8505  fzrevral  9668  nfsum1  10915  nfsum  10916  fsum2dlemstep  10993  fisumcom2  10997  bezoutlemmain  11430  setindis  12586  bdsetindis  12588