Theorem exdistr2 1962

Description: Distribution of existential quantifiers. (Contributed by NM, 17-Mar-1995.)

Assertion

Ref	Expression
exdistr2	⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦∃𝑧𝜓))

Distinct variable groups:   𝜑,𝑦   𝜑,𝑧

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem exdistr2

Step	Hyp	Ref	Expression
1		19.42vv 1961	. 2 ⊢ (∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑦∃𝑧𝜓))
2	1	exbii 1850	1 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦∃𝑧𝜓))

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:  19.42vvv  1963