Theorem 19.41vvv 1903

Description: Version of 19.41 2090 with three quantifiers and a dv 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 1902	. . 3 ⊢ (∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (∃𝑦∃𝑧𝜑 ∧ 𝜓))
2	1	exbii 1764	. 2 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ ∃𝑥(∃𝑦∃𝑧𝜑 ∧ 𝜓))
3		19.41v 1901	. 2 ⊢ (∃𝑥(∃𝑦∃𝑧𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦∃𝑧𝜑 ∧ 𝜓))
4	2, 3	bitri 263	1 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦∃𝑧𝜑 ∧ 𝜓))

Syntax hints:   ↔ wb 195   ∧ wa 383  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696

This theorem is referenced by:  19.41vvvv  1904  eloprabga  6645  dftpos3  7257