Theorem 19.41v 1895

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 1519	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
2	1	19.41h 1678	1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓))

Syntax hints:   ∧ wa 103   ↔ wb 104  ∃wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  19.41vv  1896  19.41vvv  1897  19.41vvvv  1898  exdistrv  1903  eeeanv  1926  gencbvex  2776  euxfrdc  2916  euind  2917  dfdif3  3237  r19.9rmv  3506  opabm  4265  eliunxp  4750  relop  4761  dmuni  4821  dmres  4912  dminss  5025  imainss  5026  ssrnres  5053  cnvresima  5100  resco  5115  rnco  5117  coass  5129  xpcom  5157  f11o  5475  fvelrnb  5544  rnoprab  5936  domen  6729  xpassen  6808  genpassl  7486  genpassu  7487