Theorem r19.21v 3142

Description: Restricted quantifier version of 19.21v 1940. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 2-Jan-2020.)

Assertion

Ref	Expression
r19.21v	⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem r19.21v

Step	Hyp	Ref	Expression
1		bi2.04 392	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ↔ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓)))
2	1	albii 1821	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ↔ ∀𝑥(𝜑 → (𝑥 ∈ 𝐴 → 𝜓)))
3		19.21v 1940	. . 3 ⊢ (∀𝑥(𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)))
4	2, 3	bitri 278	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)))
5		df-ral 3111	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))
6		df-ral 3111	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
7	6	imbi2i 339	. 2 ⊢ ((𝜑 → ∀𝑥 ∈ 𝐴 𝜓) ↔ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)))
8	4, 5, 7	3bitr4i 306	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   ∈ wcel 2111  ∀wral 3106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911

This theorem depends on definitions:  df-bi 210  df-ex 1782  df-ral 3111

This theorem is referenced by:  r19.23v  3238  r19.32v  3294  rmo4  3669  2reu5lem3  3696  ra4v  3814  rmo3  3818  dftr5  5139  reusv3  5271  tfinds2  7558  tfinds3  7559  wfr3g  7936  tfrlem1  7995  tfr3  8018  oeordi  8196  ordiso2  8963  ordtypelem7  8972  cantnf  9140  dfac12lem3  9556  ttukeylem5  9924  ttukeylem6  9925  fpwwe2lem8  10048  grudomon  10228  raluz2  12285  bpolycl  15398  ndvdssub  15750  gcdcllem1  15838  acsfn2  16926  pgpfac1  19195  pgpfac  19199  isdomn2  20065  islindf4  20527  isclo2  21693  1stccn  22068  kgencn  22161  txflf  22611  fclsopn  22619  nn0min  30562  bnj580  32295  bnj852  32303  rdgprc  33152  fpr3g  33235  conway  33377  filnetlem4  33842  poimirlem29  35086  heicant  35092  ntrneixb  40798  2rexrsb  43657  tfis2d  45210