Theorem exsimpl 1670

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 109	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2	1	eximi 1653	1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜑)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104  ∃wex 1545

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-ial 1587

This proof depends on definitions:  df-bi 117

This theorem is used by:  19.40  1684  euex  2116  moexexdc  2171  elex  2833  sbc5  3075  dmcoss  5052  fmptco  5874  brabvv  6134  brtpos2  6522  ringidval  14267