Theorem exsimpl 1783

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 472	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2	1	eximi 1752	1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜑)

Syntax hints:   → wi 4   ∧ wa 383  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696

This theorem is referenced by:  19.40  1785  euexALT  2499  moexex  2529  elex  3185  sbc5  3427  r19.2zb  4013  dmcoss  5306  suppimacnvss  7192  unblem2  8098  kmlem8  8862  isssc  16303  bnj1143  30115  bnj1371  30351  bnj1374  30353  bj-elissetv  32055  atex  33710  rtrclex  36943  clcnvlem  36949  pm10.55  37590