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Theorem 19.41vvvv 1954
Description: Version of 19.41 2239 with four quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by FL, 14-Jul-2007.)
Assertion
Ref Expression
19.41vvvv (∃𝑤𝑥𝑦𝑧(𝜑𝜓) ↔ (∃𝑤𝑥𝑦𝑧𝜑𝜓))
Distinct variable groups:   𝜓,𝑤   𝜓,𝑥   𝜓,𝑦   𝜓,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem 19.41vvvv
StepHypRef Expression
1 19.41vvv 1953 . . 3 (∃𝑥𝑦𝑧(𝜑𝜓) ↔ (∃𝑥𝑦𝑧𝜑𝜓))
21exbii 1849 . 2 (∃𝑤𝑥𝑦𝑧(𝜑𝜓) ↔ ∃𝑤(∃𝑥𝑦𝑧𝜑𝜓))
3 19.41v 1951 . 2 (∃𝑤(∃𝑥𝑦𝑧𝜑𝜓) ↔ (∃𝑤𝑥𝑦𝑧𝜑𝜓))
42, 3bitri 278 1 (∃𝑤𝑥𝑦𝑧(𝜑𝜓) ↔ (∃𝑤𝑥𝑦𝑧𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wex 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782
This theorem is referenced by:  elfuns  33394
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