Theorem exsimpl 1792

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 473	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2	1	eximi 1759	1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜑)

Syntax hints:   → wi 4   ∧ wa 384  ∃wex 1701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702

This theorem is referenced by:  19.40  1794  euexALT  2510  moexex  2540  elex  3198  sbc5  3442  r19.2zb  4033  dmcoss  5345  suppimacnvss  7250  unblem2  8157  kmlem8  8923  isssc  16401  bnj1143  30566  bnj1371  30802  bnj1374  30804  bj-elissetv  32505  atex  34169  rtrclex  37402  clcnvlem  37408  pm10.55  38047