Theorem exdistr2 1914

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 1911	. 2 ⊢ (∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑦∃𝑧𝜓))
2	1	exbii 1605	1 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦∃𝑧𝜓))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wex 1492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)