Theorem exdistr 1958

Description: Distribution of existential quantifiers. See also exdistrv 1959. (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 1957	. 2 ⊢ (∃𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑦𝜓))
2	1	exbii 1850	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))

Syntax hints:   ↔ wb 205   ∧ wa 396  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783

This theorem is referenced by:  exdistrv  1959  19.42vv  1961  3exdistr  1964  rexcom4  3233  sbccomlem  3803  coass  6169  uniuni  7612  eulerpartlemgvv  32343  bnj986  32935  dfiota3  34225  ac6s6f  36331