Theorem exdistr 1955

Description: Distribution of existential quantifiers. See also exdistrv 1956. (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 1954	. 2 ⊢ (∃𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑦𝜓))
2	1	exbii 1849	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))

Syntax hints:   ↔ wb 206   ∧ wa 395  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781

This theorem is referenced by:  exdistrv  1956  19.42vv  1958  3exdistr  1961  rexcom4  3259  sbccomlemOLD  3821  coass  6213  uniuni  7695  eulerpartlemgvv  34384  bnj986  34962  dfiota3  35956  ac6s6f  38212