Theorem 19.41vv 1891

Description: Theorem 19.41 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 30-Apr-1995.)

Assertion

Ref	Expression
19.41vv	⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ 𝜓))

Distinct variable groups:   𝜓,𝑥   𝜓,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 19.41vv

Step	Hyp	Ref	Expression
1		19.41v 1890	. . 3 ⊢ (∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑦𝜑 ∧ 𝜓))
2	1	exbii 1593	. 2 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(∃𝑦𝜑 ∧ 𝜓))
3		19.41v 1890	. 2 ⊢ (∃𝑥(∃𝑦𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ 𝜓))
4	2, 3	bitri 183	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ 𝜓))

Syntax hints:   ∧ wa 103   ↔ wb 104  ∃wex 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  19.41vvv  1892  rabxp  4641  rexiunxp  4746  mpomptx  5933  xpassen  6796  dmaddpqlem  7318  nqpi  7319  nqnq0pi  7379  nq0nn  7383