Theorem nfrexdxy 2351

Description: Not-free for restricted existential quantification where x and y are distinct. See nfrexdya 2353 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 2306	. 2 ⊢ (∃y ∈ A ψ ↔ ∃y(y ∈ A ∧ ψ))
2		nfraldxy.2	. . 3 ⊢ Ⅎyφ
3		nfcv 2175	. . . . . 6 ⊢ Ⅎxy
4	3	a1i 9	. . . . 5 ⊢ (φ → Ⅎxy)
5		nfraldxy.3	. . . . 5 ⊢ (φ → ℲxA)
6	4, 5	nfeld 2190	. . . 4 ⊢ (φ → Ⅎx y ∈ A)
7		nfraldxy.4	. . . 4 ⊢ (φ → Ⅎxψ)
8	6, 7	nfand 1457	. . 3 ⊢ (φ → Ⅎx(y ∈ A ∧ ψ))
9	2, 8	nfexd 1641	. 2 ⊢ (φ → Ⅎx∃y(y ∈ A ∧ ψ))
10	1, 9	nfxfrd 1361	1 ⊢ (φ → Ⅎx∃y ∈ A ψ)

Syntax hints:   → wi 4   ∧ wa 97  Ⅎwnf 1346  ∃wex 1378   ∈ wcel 1390  Ⅎwnfc 2162  ∃wrex 2301

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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306

This theorem is referenced by:  nfrexdya  2353  nfrexxy  2355  nfunid  3578  strcollnft  9444