Theorem 19.41v 1954

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 1575	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
2	1	19.41h 1733	1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wex 1541

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  19.41vv  1955  19.41vvv  1956  19.41vvvv  1957  exdistrv  1962  eeeanv  1989  gencbvex  2863  euxfrdc  3006  euind  3007  dfdif3  3333  r19.9rmv  3605  opabm  4404  eliunxp  4899  relop  4910  dmuni  4971  dmres  5064  dminss  5182  imainss  5183  ssrnres  5210  cnvresima  5257  resco  5272  rnco  5274  coass  5286  xpcom  5314  f11o  5653  fvelrnb  5729  rnoprab  6144  domen  7001  xpassen  7094  genpassl  7855  genpassu  7856