Theorem 19.42v 1962

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 1579	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2	1	19.42h 1739	1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wex 1545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  exdistr  1965  19.42vv  1967  19.42vvv  1968  4exdistr  1972  cbvex2  1978  2sb5  2043  2sb5rf  2049  rexcom4a  2846  ceqsex2  2863  reuind  3031  2rmorex  3032  sbccomlem  3126  bm1.3ii  4249  opm  4369  eqvinop  4378  uniuni  4592  elco  4941  dmopabss  4988  dmopab3  4989  mptpreima  5276  brprcneu  5683  relelfvdm  5722  fndmin  5807  fliftf  5995  dfoprab2  6125  dmoprab  6159  dmoprabss  6160  fnoprabg  6179  opabex3d  6340  opabex3  6341  eroveu  6890  dmaddpq  7736  dmmulpq  7737  prarloc  7860  ltexprlemopl  7958  ltexprlemlol  7959  ltexprlemopu  7960  ltexprlemupu  7961  shftdm  11565  fngzsum  13685  gzsumvalx  13686  ntreq0  15156  bdbm1.3ii  16831