Theorem nfrexxy 2339

Description: Not-free for restricted existential quantification where x and y are distinct. See nfrexya 2341 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 1335	. . 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 2335	. 2 ⊢ ( ⊤ → Ⅎx∃y ∈ A φ)
7	6	trud 1237	1 ⊢ Ⅎx∃y ∈ A φ

Syntax hints:   ⊤ wtru 1229  Ⅎwnf 1329  Ⅎwnfc 2147  ∃wrex 2285

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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-17 1400  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290

This theorem is referenced by:  r19.12  2400  sbcrext  2812  sbcrexg  2814  nfuni  3560  nfiunxy  3657  rexxpf  4410  abrexex2g  5670  abrexex2  5674  nfrecs  5844  bj-findis  7197  strcollnfALT  7204