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Theorem 19.23vv 1780
Description: Theorem 19.23 of [Margaris] p. 90 extended to two variables. (Contributed by NM, 10-Aug-2004.)
Assertion
Ref Expression
19.23vv (∀𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝑦𝜑𝜓))
Distinct variable groups:   𝜓,𝑥   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 19.23vv
StepHypRef Expression
1 19.23v 1779 . . 3 (∀𝑦(𝜑𝜓) ↔ (∃𝑦𝜑𝜓))
21albii 1375 . 2 (∀𝑥𝑦(𝜑𝜓) ↔ ∀𝑥(∃𝑦𝜑𝜓))
3 19.23v 1779 . 2 (∀𝑥(∃𝑦𝜑𝜓) ↔ (∃𝑥𝑦𝜑𝜓))
42, 3bitri 177 1 (∀𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝑦𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257  wex 1397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie2 1399  ax-17 1435
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  ssrel  4456  ssrelrel  4468  raliunxp  4505
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