Theorem exsimpl 1895

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 1862	1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜑)

Syntax hints:   → wi 4   ∧ wa 400  ∃wex 1806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807

This theorem is referenced by:  19.40  1913  moexexlem  2660  elissetv  2850  clelab  2913  sbc5ALT  3782  dmcoss  5963  dmcossOLD  5964  suppimacnvss  8165  unblem2  9249  kmlem8  10137  isssc  17873  krull  33702  bnj1143  35119  bnj1371  35358  bnj1374  35360  bj-sbcex  37158  atex  40065  rtrclex  44228  clcnvlem  44234  pm10.55  44964