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Theorem 19.41vv 1783
 Description: Theorem 19.41 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 30-Apr-1995.)
Assertion
Ref Expression
19.41vv (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝑦𝜑𝜓))
Distinct variable groups:   𝜓,𝑥   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 19.41vv
StepHypRef Expression
1 19.41v 1782 . . 3 (∃𝑦(𝜑𝜓) ↔ (∃𝑦𝜑𝜓))
21exbii 1496 . 2 (∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(∃𝑦𝜑𝜓))
3 19.41v 1782 . 2 (∃𝑥(∃𝑦𝜑𝜓) ↔ (∃𝑥𝑦𝜑𝜓))
42, 3bitri 173 1 (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝑦𝜑𝜓))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98  ∃wex 1381 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  19.41vvv  1784  rabxp  4380  rexiunxp  4478  mpt2mptx  5595  xpassen  6304  dmaddpqlem  6475  nqpi  6476  nqnq0pi  6536  nq0nn  6540
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