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Theorem 19.41vvvv 2029
Description: Version of 19.41 2250 with four quantifiers and a dv 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 2028 . . 3 (∃𝑥𝑦𝑧(𝜑𝜓) ↔ (∃𝑥𝑦𝑧𝜑𝜓))
21exbii 1923 . 2 (∃𝑤𝑥𝑦𝑧(𝜑𝜓) ↔ ∃𝑤(∃𝑥𝑦𝑧𝜑𝜓))
3 19.41v 2026 . 2 (∃𝑤(∃𝑥𝑦𝑧𝜑𝜓) ↔ (∃𝑤𝑥𝑦𝑧𝜑𝜓))
42, 3bitri 264 1 (∃𝑤𝑥𝑦𝑧(𝜑𝜓) ↔ (∃𝑤𝑥𝑦𝑧𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wex 1853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854
This theorem is referenced by:  elfuns  32328
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