Theorem exdistr2 1991

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 1990	. 2 ⊢ (∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑦∃𝑧𝜓))
2	1	exbii 1881	1 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦∃𝑧𝜓))

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:  19.42vvv  1992