Theorem exdistr 1950

Description: Distribution of existential quantifiers. See also exdistrv 1951. (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 1949	. 2 ⊢ (∃𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑦𝜓))
2	1	exbii 1842	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))

Syntax hints:   ↔ wb 205   ∧ wa 395  ∃wex 1773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774

This theorem is referenced by:  exdistrv  1951  19.42vv  1953  3exdistr  1956  rexcom4  3279  sbccomlem  3859  coass  6257  uniuni  7745  eulerpartlemgvv  33904  bnj986  34494  dfiota3  35427  ac6s6f  37553