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Theorem 3exbidv 1842
 Description: Formula-building rule for 3 existential quantifiers (deduction form). (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 1798 . 2 (𝜑 → (∃𝑧𝜓 ↔ ∃𝑧𝜒))
322exbidv 1841 1 (𝜑 → (∃𝑥𝑦𝑧𝜓 ↔ ∃𝑥𝑦𝑧𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∃wex 1469 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515 This theorem depends on definitions:  df-bi 116 This theorem is referenced by:  ceqsex6v  2733  euotd  4184  oprabid  5811  eloprabga  5866  eloprabi  6102
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