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Theorem 4exbidv 2006
Description: Formula-building rule for four existential quantifiers (deduction rule). (Contributed by NM, 3-Aug-1995.)
Hypothesis
Ref Expression
4exbidv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
4exbidv (𝜑 → (∃𝑥𝑦𝑧𝑤𝜓 ↔ ∃𝑥𝑦𝑧𝑤𝜒))
Distinct variable groups:   𝜑,𝑥   𝜑,𝑦   𝜑,𝑧   𝜑,𝑤
Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧,𝑤)   𝜒(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem 4exbidv
StepHypRef Expression
1 4exbidv.1 . . 3 (𝜑 → (𝜓𝜒))
212exbidv 2004 . 2 (𝜑 → (∃𝑧𝑤𝜓 ↔ ∃𝑧𝑤𝜒))
322exbidv 2004 1 (𝜑 → (∃𝑥𝑦𝑧𝑤𝜓 ↔ ∃𝑥𝑦𝑧𝑤𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wex 1852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991
This theorem depends on definitions:  df-bi 197  df-ex 1853
This theorem is referenced by:  ceqsex8v  3401  copsex4g  5086  opbrop  5338  ov3  6944  brecop  7992  addsrmo  10096  mulsrmo  10097  addsrpr  10098  mulsrpr  10099  dihopelvalcpre  37058  xihopellsmN  37064  dihopellsm  37065
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