Theorem 19.41v 1952

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  1953  19.41vvv  1954  19.41vvvv  1955  exdistrv  1960  eeeanv  1987  gencbvex  2860  euxfrdc  3002  euind  3003  dfdif3  3328  r19.9rmv  3600  opabm  4398  eliunxp  4893  relop  4904  dmuni  4965  dmres  5058  dminss  5176  imainss  5177  ssrnres  5204  cnvresima  5251  resco  5266  rnco  5268  coass  5280  xpcom  5308  f11o  5647  fvelrnb  5723  rnoprab  6135  domen  6987  xpassen  7080  genpassl  7838  genpassu  7839