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 3519 versus vtocl 3513. 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  r2al  3165  r3al  3167  ralcom4  3255  cbvraldva2  3311  ceqsralt  3471  rspc2gv  3587  elabgtOLD  3628  elabgtOLDOLD  3629  euind  3684  reu2  3685  reuind  3713  sbccomlem  3821  unissb  4890  dfiin2g  4981  axrep5  5226  asymref  6065  fvn0ssdmfun  7008  dff13  7191  mpo2eqb  7481  xpord3inddlem  8087  findcard3  9172  marypha1lem  9323  marypha2lem3  9327  aceq1  10011  kmlem15  10059  cotr2g  14883  bnj864  34889  bnj865  34890  bnj978  34916  bnj1176  34972  bnj1186  34974  dfon2lem8  35764  dffun10  35888  mpobi123f  38142  mptbi12f  38146  sn-axrep5v  42189  unielss  43191  elmapintrab  43549  undmrnresiss  43577  dfhe3  43748  dffrege115  43951  ntrneiiso  44064  ntrneikb  44067  pm10.541  44340  pm10.542  44341  19.21vv  44349  pm11.62  44367  2sbc6g  44388  2rexsb  47085