Theorem exsimpl 1887

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 486	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2	1	eximi 1854	1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜑)

Syntax hints:   → wi 4   ∧ wa 399  ∃wex 1798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799

This theorem is referenced by:  19.40  1905  moexexlem  2652  elissetv  2842  clelab  2905  sbc5ALT  3771  dmcoss  5947  dmcossOLD  5948  suppimacnvss  8147  unblem2  9231  kmlem8  10108  isssc  17844  krull  33628  bnj1143  35046  bnj1371  35285  bnj1374  35287  bj-sbcex  37084  atex  39991  rtrclex  44154  clcnvlem  44160  pm10.55  44906