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 3527 versus vtocl 3521. 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  3171  r3al  3173  ralcom4  3261  cbvraldva2  3318  ceqsralt  3479  rspc2gv  3595  elabgtOLD  3636  elabgtOLDOLD  3637  euind  3692  reu2  3693  reuind  3721  sbccomlem  3829  unissb  4899  unissbOLD  4900  dfiin2g  4991  axrep5  5237  asymref  6077  fvn0ssdmfun  7028  dff13  7211  mpo2eqb  7501  xpord3inddlem  8110  findcard3  9205  findcard3OLD  9206  marypha1lem  9360  marypha2lem3  9364  aceq1  10046  kmlem15  10094  cotr2g  14918  bnj864  34885  bnj865  34886  bnj978  34912  bnj1176  34968  bnj1186  34970  dfon2lem8  35751  dffun10  35875  mpobi123f  38129  mptbi12f  38133  sn-axrep5v  42177  unielss  43180  elmapintrab  43538  undmrnresiss  43566  dfhe3  43737  dffrege115  43940  ntrneiiso  44053  ntrneikb  44056  pm10.541  44329  pm10.542  44330  19.21vv  44338  pm11.62  44356  2sbc6g  44377  2rexsb  47075