Theorem 19.42vv 1967

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 1965	. 2 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
2		19.42v 1962	. 2 ⊢ (∃𝑥(𝜑 ∧ ∃𝑦𝜓) ↔ (𝜑 ∧ ∃𝑥∃𝑦𝜓))
3	1, 2	bitri 184	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥∃𝑦𝜓))

This proof depends on syntax axioms:   ∧ wa 104   ↔ wb 105  ∃wex 1545

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587

This proof depends on definitions:  df-bi 117

This theorem is used by:  19.42vvv  1968  19.42vvvv  1969  exdistr2  1970  3exdistr  1971  ceqsex3v  2865  ceqsex4v  2866  elvvv  4838  dfoprab2  6135  resoprab  6184  ovi3  6226  ov6g  6227  oprabex3  6362  xpassen  7128  enq0enq  7798  enq0sym  7799  nqnq0pi  7805  axaddf  8235  axmulf  8236