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Theorem 3exbidv 2004
Description: Formula-building rule for three existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
Hypothesis
Ref Expression
3exbidv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
3exbidv (𝜑 → (∃𝑥𝑦𝑧𝜓 ↔ ∃𝑥𝑦𝑧𝜒))
Distinct variable groups:   𝜑,𝑥   𝜑,𝑦   𝜑,𝑧
Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧)   𝜒(𝑥,𝑦,𝑧)

Proof of Theorem 3exbidv
StepHypRef Expression
1 3exbidv.1 . . 3 (𝜑 → (𝜓𝜒))
21exbidv 2001 . 2 (𝜑 → (∃𝑧𝜓 ↔ ∃𝑧𝜒))
322exbidv 2003 1 (𝜑 → (∃𝑥𝑦𝑧𝜓 ↔ ∃𝑥𝑦𝑧𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wex 1851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990
This theorem depends on definitions:  df-bi 197  df-ex 1852
This theorem is referenced by:  ceqsex6v  3397  euotd  5106  oprabid  6821  eloprabga  6893  eloprabi  7381  bnj981  31352
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