Theorem exdistr 1906

Description: Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)

Assertion

Ref	Expression
exdistr	⊢ (∃x∃y(φ ∧ ψ) ↔ ∃x(φ ∧ ∃yψ))

Distinct variable group:   φ,y

Allowed substitution hints:   φ(x)   ψ(x, y)

Proof of Theorem exdistr

Step	Hyp	Ref	Expression
1		19.42v 1905	. 2 ⊢ (∃y(φ ∧ ψ) ↔ (φ ∧ ∃yψ))
2	1	exbii 1582	1 ⊢ (∃x∃y(φ ∧ ψ) ↔ ∃x(φ ∧ ∃yψ))

This proof depends on syntax axioms:   ↔ wb 176   ∧ wa 358  ∃wex 1541

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545

This theorem is used by:  19.42vv  1907  3exdistr  1910  sbel2x  2125  sbccomlem  3117  otkelins3kg  4255  el1st  4730  elres  4996