Theorem exdistr2 1907

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 1904	. 2 ⊢ (∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑦∃𝑧𝜓))
2	1	exbii 1598	1 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦∃𝑧𝜓))

Syntax hints:   ∧ wa 103   ↔ wb 104  ∃wex 1485

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)