Theorem exsimpl 1901

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 488	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2	1	eximi 1868	1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜑)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by:  19.40  1919  moexexlem  2653  elissetv  2843  clelab  2906  sbc5ALT  3771  dmcoss  5963  dmcossOLD  5964  suppimacnvss  8175  unblem2  9267  kmlem8  10164  isssc  17915  krull  33889  bnj1143  35307  bnj1371  35546  bnj1374  35548  bj-sbcex  37389  atex  40287  rtrclex  44465  clcnvlem  44471  pm10.55  45201