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  21623  islindf4  21780  ovolgelb  25414  dyadmax  25532  itg2leub  25668  eqscut2  27752  addsprop  27923  addsuniflem  27948  negsprop  27981  mulsprop  28073  mulsuniflem  28092  mptelee  28875  nmoubi  30751  nmopub  31887  nmfnleub  31904  sigaclcu2  34103  untuni  35689  elintfv  35745  heibor1lem  37796  ispsubsp2  39733  pmapglbx  39756  neik0pk1imk0  44029  2reuimp0  47108