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Theorem eean 2349
Description: Distribute existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypotheses
Ref Expression
eean.1 𝑦𝜑
eean.2 𝑥𝜓
Assertion
Ref Expression
eean (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

Proof of Theorem eean
StepHypRef Expression
1 eean.1 . . . 4 𝑦𝜑
2119.42 2234 . . 3 (∃𝑦(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑦𝜓))
32exbii 1845 . 2 (∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
4 eean.2 . . . 4 𝑥𝜓
54nfex 2323 . . 3 𝑥𝑦𝜓
6519.41 2233 . 2 (∃𝑥(𝜑 ∧ ∃𝑦𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
73, 6bitri 275 1 (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wex 1776  wnf 1780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-11 2155  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781
This theorem is referenced by:  eeanv  2350  reean  3314
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