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Theorem 4exbidv 1930
Description: Formula-building rule for four existential quantifiers (deduction form). (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 1928 . 2 (𝜑 → (∃𝑧𝑤𝜓 ↔ ∃𝑧𝑤𝜒))
322exbidv 1928 1 (𝜑 → (∃𝑥𝑦𝑧𝑤𝜓 ↔ ∃𝑥𝑦𝑧𝑤𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wex 1782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914
This theorem depends on definitions:  df-bi 206  df-ex 1783
This theorem is referenced by:  ceqsex8v  3533  copsex4g  5493  opbrop  5770  ov3  7564  brecop  8799  addsrmo  11063  mulsrmo  11064  addsrpr  11065  mulsrpr  11066  dihopelvalcpre  40056  xihopellsmN  40062  dihopellsm  40063
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