Theorem nfrexdxy 2504

Description: Not-free for restricted existential quantification where 𝑥 and 𝑦 are distinct. See nfrexdya 2506 for a version with 𝑦 and 𝐴 distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)

Hypotheses

Ref	Expression
nfraldxy.2	⊢ Ⅎ𝑦𝜑
nfraldxy.3	⊢ (𝜑 → Ⅎ𝑥𝐴)
nfraldxy.4	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfrexdxy	⊢ (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem nfrexdxy

Step	Hyp	Ref	Expression
1		df-rex 2454	. 2 ⊢ (∃𝑦 ∈ 𝐴 𝜓 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
2		nfraldxy.2	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfcv 2312	. . . . . 6 ⊢ Ⅎ𝑥𝑦
4	3	a1i 9	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝑦)
5		nfraldxy.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴)
6	4, 5	nfeld 2328	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
7		nfraldxy.4	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
8	6, 7	nfand 1561	. . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓))
9	2, 8	nfexd 1754	. 2 ⊢ (𝜑 → Ⅎ𝑥∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
10	1, 9	nfxfrd 1468	1 ⊢ (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∧ wa 103  Ⅎwnf 1453  ∃wex 1485   ∈ wcel 2141  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-nf 1454  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454

This theorem is referenced by:  nfrexdya  2506  nfrexxy  2509  nfunid  3803  strcollnft  14019