Theorem 19.41vvvv 1985

Description: Version of 19.41 2274 with four quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by FL, 14-Jul-2007.)

Assertion

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

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

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

Proof of Theorem 19.41vvvv

Step	Hyp	Ref	Expression
1		19.41vvv 1984	. . 3 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦∃𝑧𝜑 ∧ 𝜓))
2	1	exbii 1881	. 2 ⊢ (∃𝑤∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ ∃𝑤(∃𝑥∃𝑦∃𝑧𝜑 ∧ 𝜓))
3		19.41v 1982	. 2 ⊢ (∃𝑤(∃𝑥∃𝑦∃𝑧𝜑 ∧ 𝜓) ↔ (∃𝑤∃𝑥∃𝑦∃𝑧𝜑 ∧ 𝜓))
4	2, 3	bitri 278	1 ⊢ (∃𝑤∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (∃𝑤∃𝑥∃𝑦∃𝑧𝜑 ∧ 𝜓))

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:  elfuns  36418