Theorem exdistr 1897

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 1894	. 2 ⊢ (∃𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑦𝜓))
2	1	exbii 1593	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))

Syntax hints:   ∧ wa 103   ↔ wb 104  ∃wex 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  exdistrv  1898  19.42vv  1899  3exdistr  1903  sbel2x  1986  sbexyz  1991  sbccomlem  3025  uniuni  4429  coass  5122  subhalfnqq  7355