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Theorem 3exbidv 1749
 Description: Formula-building rule for 3 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 1706 . 2 (𝜑 → (∃𝑧𝜓 ↔ ∃𝑧𝜒))
322exbidv 1748 1 (𝜑 → (∃𝑥𝑦𝑧𝜓 ↔ ∃𝑥𝑦𝑧𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∃wex 1381 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  ceqsex6v  2598  euotd  3991  oprabid  5537  eloprabga  5591  eloprabi  5822
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