Theorem 19.41vvv 1983

Description: Version of 19.41 2271 with three quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by NM, 30-Apr-1995.)

Assertion

Ref	Expression
19.41vvv	⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦∃𝑧𝜑 ∧ 𝜓))

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

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

Proof of Theorem 19.41vvv

Step	Hyp	Ref	Expression
1		19.41vv 1982	. . 3 ⊢ (∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (∃𝑦∃𝑧𝜑 ∧ 𝜓))
2	1	exbii 1880	. 2 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ ∃𝑥(∃𝑦∃𝑧𝜑 ∧ 𝜓))
3		19.41v 1981	. 2 ⊢ (∃𝑥(∃𝑦∃𝑧𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦∃𝑧𝜑 ∧ 𝜓))
4	2, 3	bitri 278	1 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦∃𝑧𝜑 ∧ 𝜓))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812

This theorem is used by:  19.41vvvv  1984  eloprabga  7531  dftpos3  8266