Theorem r19.23v 3189

Description: Restricted quantifier version of 19.23v 1962. Version of r19.23 3259 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 318	. . 3 ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜓 → ¬ 𝜑))
2	1	ralbii 3108	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥 ∈ 𝐴 (¬ 𝜓 → ¬ 𝜑))
3		r19.21v 3187	. 2 ⊢ (∀𝑥 ∈ 𝐴 (¬ 𝜓 → ¬ 𝜑) ↔ (¬ 𝜓 → ∀𝑥 ∈ 𝐴 ¬ 𝜑))
4		dfrex2 3089	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)
5	4	imbi1i 351	. . 3 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 → 𝜓) ↔ (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 → 𝜓))
6		con1b 360	. . 3 ⊢ ((¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 → 𝜓) ↔ (¬ 𝜓 → ∀𝑥 ∈ 𝐴 ¬ 𝜑))
7	5, 6	bitr2i 278	. 2 ⊢ ((¬ 𝜓 → ∀𝑥 ∈ 𝐴 ¬ 𝜑) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓))
8	2, 3, 7	3bitri 299	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208  ∀wral 3076  ∃wrex 3086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800  df-ral 3077  df-rex 3087

This theorem is referenced by:  ceqsralv  3494  ralxpxfr2d  3605  uniiunlem  4040  2reu4lem  4477  dfiin2g  4988  iunss  5002  iunssOLD  5003  replem  5238  ralxfr2d  5367  ssrel2  5757  idrefALT  6100  dfpo2  6283  funimass4  6931  fnssintima  7346  ralrnmpo  7535  imaeqalov  7635  ttrclss  9675  kmlem12  10118  fimaxre3  12138  gcdcllem1  16533  vdwmc2  17015  iunocv  21733  islindf4  21890  ovolgelb  25542  dyadmax  25660  itg2leub  25796  eqcuts2  27879  addsprop  28069  addsuniflem  28094  negsprop  28128  mulsprop  28223  mulsuniflem  28242  mpteleeOLD  29096  nmoubi  30975  nmopub  32111  nmfnleub  32128  sigaclcu2  34417  untuni  36059  elintfv  36115  heibor1lem  38308  ispsubsp2  40370  pmapglbx  40393  neik0pk1imk0  44623  2reuimp0  47708