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Theorem r3ex 3196
Description: Triple existential quantification. (Contributed by AV, 21-Jul-2025.)
Assertion
Ref Expression
r3ex (∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∃𝑥𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) ∧ 𝜑))
Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝐴,𝑧   𝑧,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦,𝑧)

Proof of Theorem r3ex
StepHypRef Expression
1 r2ex 3194 . 2 (∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ ∃𝑧𝐶 𝜑))
2 df-rex 3069 . . . . 5 (∃𝑧𝐶 𝜑 ↔ ∃𝑧(𝑧𝐶𝜑))
32anbi2i 623 . . . 4 (((𝑥𝐴𝑦𝐵) ∧ ∃𝑧𝐶 𝜑) ↔ ((𝑥𝐴𝑦𝐵) ∧ ∃𝑧(𝑧𝐶𝜑)))
4 19.42v 1951 . . . 4 (∃𝑧((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐶𝜑)) ↔ ((𝑥𝐴𝑦𝐵) ∧ ∃𝑧(𝑧𝐶𝜑)))
5 anass 468 . . . . . . 7 ((((𝑥𝐴𝑦𝐵) ∧ 𝑧𝐶) ∧ 𝜑) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐶𝜑)))
65bicomi 224 . . . . . 6 (((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐶𝜑)) ↔ (((𝑥𝐴𝑦𝐵) ∧ 𝑧𝐶) ∧ 𝜑))
7 df-3an 1088 . . . . . . 7 ((𝑥𝐴𝑦𝐵𝑧𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧𝐶))
87bicomi 224 . . . . . 6 (((𝑥𝐴𝑦𝐵) ∧ 𝑧𝐶) ↔ (𝑥𝐴𝑦𝐵𝑧𝐶))
96, 8bianbi 627 . . . . 5 (((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐶𝜑)) ↔ ((𝑥𝐴𝑦𝐵𝑧𝐶) ∧ 𝜑))
109exbii 1845 . . . 4 (∃𝑧((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐶𝜑)) ↔ ∃𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) ∧ 𝜑))
113, 4, 103bitr2i 299 . . 3 (((𝑥𝐴𝑦𝐵) ∧ ∃𝑧𝐶 𝜑) ↔ ∃𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) ∧ 𝜑))
12112exbii 1846 . 2 (∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ ∃𝑧𝐶 𝜑) ↔ ∃𝑥𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) ∧ 𝜑))
131, 12bitri 275 1 (∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∃𝑥𝑦𝑧((𝑥𝐴𝑦𝐵𝑧𝐶) ∧ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1086  wex 1776  wcel 2106  wrex 3068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-ex 1777  df-ral 3060  df-rex 3069
This theorem is referenced by:  hash3tpb  14531
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