Theorem exsimpl 1944

Description: Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
exsimpl	⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜑)

Proof of Theorem exsimpl

Step	Hyp	Ref	Expression
1		simpl 474	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2	1	eximi 1911	1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜑)

Syntax hints:   → wi 4   ∧ wa 383  ∃wex 1853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886

This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854

This theorem is referenced by:  19.40  1946  euexALT  2649  moexex  2679  elex  3352  sbc5  3601  r19.2zb  4205  dmcoss  5540  suppimacnvss  7474  unblem2  8380  kmlem8  9191  isssc  16701  bnj1143  31189  bnj1371  31425  bnj1374  31427  bj-elissetv  33183  atex  35213  rtrclex  38444  clcnvlem  38450  pm10.55  39088