Theorem nfrexw 2581

Description: Not-free for restricted existential quantification where 𝑥 and 𝑦 are distinct. See nfrexya 2583 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
nfrexw	⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem nfrexw

Step	Hyp	Ref	Expression
1		nftru 1515	. . 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 2576	. 2 ⊢ (⊤ → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑)
7	6	mptru 1407	1 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑

Syntax hints:  ⊤wtru 1399  Ⅎwnf 1509  Ⅎwnfc 2371  ∃wrex 2521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526

This theorem is referenced by:  r19.12  2649  sbcrext  3119  nfuni  3919  nfiunxy  4016  rexxpf  4901  abrexex2g  6312  abrexex2  6316  nfrecs  6537  nfwrd  11249  fimaxre2  11908  nfsum  12038  nfcprod1  12236  nfcprod  12237  bezoutlemmain  12690  ctiunctlemfo  13182  bj-findis  16741  strcollnfALT  16748