Theorem 19.42vv 1836

Description: Theorem 19.42 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 16-Mar-1995.)

Assertion

Ref	Expression
19.42vv	⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥∃𝑦𝜓))

Distinct variable groups:   𝜑,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜓(𝑥,𝑦)

Proof of Theorem 19.42vv

Step	Hyp	Ref	Expression
1		exdistr 1835	. 2 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
2		19.42v 1834	. 2 ⊢ (∃𝑥(𝜑 ∧ ∃𝑦𝜓) ↔ (𝜑 ∧ ∃𝑥∃𝑦𝜓))
3	1, 2	bitri 182	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥∃𝑦𝜓))

Syntax hints:   ∧ wa 102   ↔ wb 103  ∃wex 1426

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  19.42vvv  1837  19.42vvvv  1838  exdistr2  1839  3exdistr  1840  ceqsex3v  2661  ceqsex4v  2662  elvvv  4501  dfoprab2  5696  resoprab  5741  ovi3  5781  ov6g  5782  oprabex3  5900  xpassen  6546  enq0enq  6990  enq0sym  6991  nqnq0pi  6997