Theorem 19.41vvv 1919

Description: Theorem 19.41 of [Margaris] p. 90 with 3 quantifiers. (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 1918	. . 3 ⊢ (∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (∃𝑦∃𝑧𝜑 ∧ 𝜓))
2	1	exbii 1619	. 2 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ ∃𝑥(∃𝑦∃𝑧𝜑 ∧ 𝜓))
3		19.41v 1917	. 2 ⊢ (∃𝑥(∃𝑦∃𝑧𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦∃𝑧𝜑 ∧ 𝜓))
4	2, 3	bitri 184	1 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦∃𝑧𝜑 ∧ 𝜓))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wex 1506

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  19.41vvvv  1920  eloprabga  6013  dftpos3  6329