Theorem 19.42vvvv 1923

Description: Theorem 19.42 of [Margaris] p. 90 with 4 quantifiers. (Contributed by Jim Kingdon, 23-Nov-2019.)

Assertion

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

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

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

Proof of Theorem 19.42vvvv

Step	Hyp	Ref	Expression
1		19.42vv 1921	. . 3 ⊢ (∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑦∃𝑧𝜓))
2	1	2exbii 1616	. 2 ⊢ (∃𝑤∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ ∃𝑤∃𝑥(𝜑 ∧ ∃𝑦∃𝑧𝜓))
3		19.42vv 1921	. 2 ⊢ (∃𝑤∃𝑥(𝜑 ∧ ∃𝑦∃𝑧𝜓) ↔ (𝜑 ∧ ∃𝑤∃𝑥∃𝑦∃𝑧𝜓))
4	2, 3	bitri 184	1 ⊢ (∃𝑤∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑤∃𝑥∃𝑦∃𝑧𝜓))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wex 1502

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 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-4 1520  ax-17 1536  ax-ial 1544

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  ceqsex8v  2794  enq0tr  7446