Theorem 19.42v 1835

Description: Special case of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
19.42v	⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem 19.42v

Step	Hyp	Ref	Expression
1		ax-17 1465	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2	1	19.42h 1623	1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))

Syntax hints:   ∧ wa 103   ↔ wb 104  ∃wex 1427

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-4 1446  ax-17 1465  ax-ial 1473

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  exdistr  1836  19.42vv  1837  19.42vvv  1838  4exdistr  1842  cbvex2  1846  2sb5  1908  2sb5rf  1914  rexcom4a  2644  ceqsex2  2660  reuind  2821  2rmorex  2822  sbccomlem  2914  bm1.3ii  3966  opm  4070  eqvinop  4079  uniuni  4286  elco  4615  dmopabss  4661  dmopab3  4662  mptpreima  4937  brprcneu  5311  relelfvdm  5349  fndmin  5420  fliftf  5592  dfoprab2  5710  dmoprab  5743  dmoprabss  5744  fnoprabg  5760  opabex3d  5906  opabex3  5907  eroveu  6397  dmaddpq  6999  dmmulpq  7000  prarloc  7123  ltexprlemopl  7221  ltexprlemlol  7222  ltexprlemopu  7223  ltexprlemupu  7224  shftdm  10317  ntreq0  11893  bdbm1.3ii  12055