Theorem r19.23v 3160

Description: Restricted quantifier version of 19.23v 1942. Version of r19.23 3232 with a disjoint variable condition. (Contributed by NM, 31-Aug-1999.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2020.)

Assertion

Ref	Expression
r19.23v	⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓))

Distinct variable group:   𝜓,𝑥

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

Proof of Theorem r19.23v

Step	Hyp	Ref	Expression
1		con34b 316	. . 3 ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜓 → ¬ 𝜑))
2	1	ralbii 3075	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥 ∈ 𝐴 (¬ 𝜓 → ¬ 𝜑))
3		r19.21v 3158	. 2 ⊢ (∀𝑥 ∈ 𝐴 (¬ 𝜓 → ¬ 𝜑) ↔ (¬ 𝜓 → ∀𝑥 ∈ 𝐴 ¬ 𝜑))
4		dfrex2 3056	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)
5	4	imbi1i 349	. . 3 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 → 𝜓) ↔ (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 → 𝜓))
6		con1b 358	. . 3 ⊢ ((¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 → 𝜓) ↔ (¬ 𝜓 → ∀𝑥 ∈ 𝐴 ¬ 𝜑))
7	5, 6	bitr2i 276	. 2 ⊢ ((¬ 𝜓 → ∀𝑥 ∈ 𝐴 ¬ 𝜑) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓))
8	2, 3, 7	3bitri 297	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wral 3044  ∃wrex 3053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-ral 3045  df-rex 3054

This theorem is referenced by:  ceqsralv  3485  ralxpxfr2d  3609  uniiunlem  4046  2reu4lem  4481  dfiin2g  4991  iunss  5004  ralxfr2d  5360  ssrel2  5739  reliun  5770  idrefALT  6072  dfpo2  6257  funimass4  6907  fnssintima  7319  ralrnmpo  7508  imaeqalov  7608  ttrclss  9649  kmlem12  10091  fimaxre3  12105  gcdcllem1  16445  vdwmc2  16926  iunocv  21566  islindf4  21723  ovolgelb  25357  dyadmax  25475  itg2leub  25611  eqscut2  27694  addsprop  27859  addsuniflem  27884  negsprop  27917  mulsprop  28009  mulsuniflem  28028  mptelee  28798  nmoubi  30674  nmopub  31810  nmfnleub  31827  sigaclcu2  34083  untuni  35669  elintfv  35725  heibor1lem  37776  ispsubsp2  39713  pmapglbx  39736  neik0pk1imk0  44009  2reuimp0  47088