Theorem exsimpl 1898

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 487	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2	1	eximi 1865	1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜑)

Syntax hints:   → wi 4   ∧ wa 400  ∃wex 1809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810

This theorem is referenced by:  19.40  1916  moexexlem  2654  elissetv  2844  clelab  2907  sbc5ALT  3774  dmcoss  5967  dmcossOLD  5968  suppimacnvss  8170  unblem2  9254  kmlem8  10142  isssc  17878  krull  33742  bnj1143  35159  bnj1371  35398  bnj1374  35400  bj-sbcex  37254  atex  40161  rtrclex  44326  clcnvlem  44332  pm10.55  45062