Theorem nfriotadxy 5419

Description: Deduction version of nfriota 5420. (Contributed by Jim Kingdon, 12-Jan-2019.)

Hypotheses

Ref	Expression
nfriotadxy.1	⊢ Ⅎyφ
nfriotadxy.2	⊢ (φ → Ⅎxψ)
nfriotadxy.3	⊢ (φ → ℲxA)

Assertion

Ref	Expression
nfriotadxy	⊢ (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 nfriotadxy

Step	Hyp	Ref	Expression
1		df-riota 5411	. 2 ⊢ (℩y ∈ A ψ) = (℩y(y ∈ A ∧ ψ))
2		nfriotadxy.1	. . 3 ⊢ Ⅎyφ
3		nfcv 2175	. . . . . 6 ⊢ Ⅎxy
4	3	a1i 9	. . . . 5 ⊢ (φ → Ⅎxy)
5		nfriotadxy.3	. . . . 5 ⊢ (φ → ℲxA)
6	4, 5	nfeld 2190	. . . 4 ⊢ (φ → Ⅎx y ∈ A)
7		nfriotadxy.2	. . . 4 ⊢ (φ → Ⅎxψ)
8	6, 7	nfand 1457	. . 3 ⊢ (φ → Ⅎx(y ∈ A ∧ ψ))
9	2, 8	nfiotadxy 4813	. 2 ⊢ (φ → Ⅎx(℩y(y ∈ A ∧ ψ)))
10	1, 9	nfcxfrd 2173	1 ⊢ (φ → Ⅎx(℩y ∈ A ψ))

Syntax hints:   → wi 4   ∧ wa 97  Ⅎwnf 1346   ∈ wcel 1390  Ⅎwnfc 2162  ℩cio 4808  ℩crio 5410

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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-sn 3373  df-uni 3572  df-iota 4810  df-riota 5411

This theorem is referenced by:  nfriota  5420