Theorem nfrexxy 2509

Description: Not-free for restricted existential quantification where 𝑥 and 𝑦 are distinct. See nfrexya 2511 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 1459	. . 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 2504	. 2 ⊢ (⊤ → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑)
7	6	mptru 1357	1 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑

Syntax hints:  ⊤wtru 1349  Ⅎwnf 1453  Ⅎwnfc 2299  ∃wrex 2449

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454

This theorem is referenced by:  r19.12  2576  sbcrext  3032  nfuni  3802  nfiunxy  3899  rexxpf  4758  abrexex2g  6099  abrexex2  6103  nfrecs  6286  fimaxre2  11190  nfsum  11320  nfcprod1  11517  nfcprod  11518  bezoutlemmain  11953  ctiunctlemfo  12394  bj-findis  14014  strcollnfALT  14021