Theorem 19.41v 1949

Description: Special case of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
19.41v	⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓))

Distinct variable group:   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem 19.41v

Step	Hyp	Ref	Expression
1		ax-17 1572	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
2	1	19.41h 1731	1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wex 1538

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  19.41vv  1950  19.41vvv  1951  19.41vvvv  1952  exdistrv  1957  eeeanv  1984  gencbvex  2847  euxfrdc  2989  euind  2990  dfdif3  3314  r19.9rmv  3583  opabm  4368  eliunxp  4858  relop  4869  dmuni  4930  dmres  5022  dminss  5139  imainss  5140  ssrnres  5167  cnvresima  5214  resco  5229  rnco  5231  coass  5243  xpcom  5271  f11o  5601  fvelrnb  5674  rnoprab  6078  domen  6890  xpassen  6977  genpassl  7699  genpassu  7700