Theorem 19.41v 1951

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  1952  19.41vvv  1953  19.41vvvv  1954  exdistrv  1959  eeeanv  1986  gencbvex  2851  euxfrdc  2993  euind  2994  dfdif3  3319  r19.9rmv  3588  opabm  4381  eliunxp  4875  relop  4886  dmuni  4947  dmres  5040  dminss  5158  imainss  5159  ssrnres  5186  cnvresima  5233  resco  5248  rnco  5250  coass  5262  xpcom  5290  f11o  5626  fvelrnb  5702  rnoprab  6114  domen  6965  xpassen  7057  genpassl  7787  genpassu  7788