Theorem nfrexxy 2505

Description: Not-free for restricted existential quantification where 𝑥 and 𝑦 are distinct. See nfrexya 2507 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 1454	. . 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 2500	. 2 ⊢ (⊤ → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑)
7	6	mptru 1352	1 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑

Syntax hints:  ⊤wtru 1344  Ⅎwnf 1448  Ⅎwnfc 2295  ∃wrex 2445

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450

This theorem is referenced by:  r19.12  2572  sbcrext  3028  nfuni  3795  nfiunxy  3892  rexxpf  4751  abrexex2g  6088  abrexex2  6092  nfrecs  6275  fimaxre2  11168  nfsum  11298  nfcprod1  11495  nfcprod  11496  bezoutlemmain  11931  ctiunctlemfo  12372  bj-findis  13861  strcollnfALT  13868