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  3505  opabm  4263  eliunxp  4748  relop  4759  dmuni  4819  dmres  4910  dminss  5023  imainss  5024  ssrnres  5051  cnvresima  5098  resco  5113  rnco  5115  coass  5127  xpcom  5155  f11o  5473  fvelrnb  5542  rnoprab  5933  domen  6725  xpassen  6804  genpassl  7473  genpassu  7474