Theorem nfrexxy 2355

Description: Not-free for restricted existential quantification where x and y are distinct. See nfrexya 2357 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
nfrexxy	⊢ Ⅎx∃y ∈ A φ

Distinct variable group:   x,y

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

Proof of Theorem nfrexxy

Step	Hyp	Ref	Expression
1		nftru 1352	. . 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	nfrexdxy 2351	. 2 ⊢ ( ⊤ → Ⅎx∃y ∈ A φ)
7	6	trud 1251	1 ⊢ Ⅎx∃y ∈ A φ

Syntax hints:   ⊤ wtru 1243  Ⅎwnf 1346  Ⅎwnfc 2162  ∃wrex 2301

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306

This theorem is referenced by:  r19.12  2416  sbcrext  2829  sbcrexg  2831  nfuni  3577  nfiunxy  3674  rexxpf  4426  abrexex2g  5689  abrexex2  5693  nfrecs  5863  bj-findis  9409  strcollnfALT  9416