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  4370  eliunxp  4864  relop  4875  dmuni  4936  dmres  5029  dminss  5146  imainss  5147  ssrnres  5174  cnvresima  5221  resco  5236  rnco  5238  coass  5250  xpcom  5278  f11o  5610  fvelrnb  5686  rnoprab  6096  domen  6913  xpassen  7002  genpassl  7727  genpassu  7728