Theorem exdistr 1987

Description: Distribution of existential quantifiers. See also exdistrv 1988. (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 1986	. 2 ⊢ (∃𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑦𝜓))
2	1	exbii 1881	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by:  exdistrv  1988  19.42vv  1990  3exdistr  1993  rexcom4  3295  sbccomlemOLD  3826  coass  6272  uniuni  7770  eulerpartlemgvv  34798  bnj986  35375  dfiota3  36434  ac6s6f  38863