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  2656  elissetv  2846  clelab  2909  sbc5ALT  3775  dmcoss  5967  dmcossOLD  5968  suppimacnvss  8171  unblem2  9256  kmlem8  10153  isssc  17894  krull  33784  bnj1143  35202  bnj1371  35441  bnj1374  35443  bj-sbcex  37306  atex  40213  rtrclex  44376  clcnvlem  44382  pm10.55  45112