Theorem 19.21v 1939

Description: Version of 19.21 2208 with a disjoint variable condition, requiring fewer axioms.

Notational convention: We sometimes suffix with "v" the label of a theorem using a distinct variable ("dv") condition instead of a nonfreeness hypothesis such as Ⅎ𝑥𝜑. Conversely, we sometimes suffix with "f" the label of a theorem introducing such a nonfreeness hypothesis ("f" stands for "not free in", see df-nf 1784) instead of a disjoint variable condition. For instance, 19.21v 1939 versus 19.21 2208 and vtoclf 3530 versus vtocl 3524. Note that "not free in" is less restrictive than "does not occur in". Note that the version with a disjoint variable condition is easily proved from the version with the corresponding nonfreeness hypothesis, by using nfv 1914. However, the dv version can often be proved from fewer axioms. (Contributed by NM, 21-Jun-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 2-Jan-2020.) (Proof shortened by Wolf Lammen, 12-Jul-2020.)

Assertion

Ref	Expression
19.21v	⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem 19.21v

Step	Hyp	Ref	Expression
1		stdpc5v 1938	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))
2		ax5e 1912	. . . 4 ⊢ (∃𝑥𝜑 → 𝜑)
3	2	imim1i 63	. . 3 ⊢ ((𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))
4		19.38 1839	. . 3 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))
5	3, 4	syl 17	. 2 ⊢ ((𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))
6	1, 5	impbii 209	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538  ∃wex 1779

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-ex 1780

This theorem is referenced by:  19.32v  1940  pm11.53v  1944  19.12vvv  1994  cbvaldvaw  2038  2sb6  2087  sbrimvw  2092  sbal  2170  hbsbwOLD  2173  sbrim  2304  pm11.53  2344  19.12vv  2345  sbhb  2519  r19.21vOLD  3159  r2al  3173  r3al  3175  ralcom4  3263  cbvraldva2  3321  ceqsralt  3482  rspc2gv  3598  elabgtOLD  3639  elabgtOLDOLD  3640  euind  3695  reu2  3696  reuind  3724  sbccomlem  3832  unissb  4903  unissbOLD  4904  dfiin2g  4996  axrep5  5242  asymref  6089  fvn0ssdmfun  7046  dff13  7229  mpo2eqb  7521  xpord3inddlem  8133  findcard3  9229  findcard3OLD  9230  marypha1lem  9384  marypha2lem3  9388  aceq1  10070  kmlem15  10118  cotr2g  14942  bnj864  34912  bnj865  34913  bnj978  34939  bnj1176  34995  bnj1186  34997  dfon2lem8  35778  dffun10  35902  mpobi123f  38156  mptbi12f  38160  sn-axrep5v  42204  unielss  43207  elmapintrab  43565  undmrnresiss  43593  dfhe3  43764  dffrege115  43967  ntrneiiso  44080  ntrneikb  44083  pm10.541  44356  pm10.542  44357  19.21vv  44365  pm11.62  44383  2sbc6g  44404  2rexsb  47102