Theorem 19.42vvv 1956

Description: Version of 19.42 2234 with three quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by NM, 21-Sep-2011.) (Proof shortened by Wolf Lammen, 27-Aug-2023.)

Assertion

Ref	Expression
19.42vvv	⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥∃𝑦∃𝑧𝜓))

Distinct variable groups:   𝜑,𝑥   𝜑,𝑦   𝜑,𝑧

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

Proof of Theorem 19.42vvv

Step	Hyp	Ref	Expression
1		exdistr2 1955	. 2 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦∃𝑧𝜓))
2		19.42v 1950	. 2 ⊢ (∃𝑥(𝜑 ∧ ∃𝑦∃𝑧𝜓) ↔ (𝜑 ∧ ∃𝑥∃𝑦∃𝑧𝜓))
3	1, 2	bitri 277	1 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥∃𝑦∃𝑧𝜓))

Syntax hints:   ↔ wb 208   ∧ wa 398  ∃wex 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777

This theorem is referenced by:  ceqsex6v  3547