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Theorem 2exsb 2021
Description: An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.)
Assertion
Ref Expression
2exsb (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝑤,𝑧   𝜑,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2exsb
StepHypRef Expression
1 exsb 2020 . . . 4 (∃𝑦𝜑 ↔ ∃𝑤𝑦(𝑦 = 𝑤𝜑))
21exbii 1616 . . 3 (∃𝑥𝑦𝜑 ↔ ∃𝑥𝑤𝑦(𝑦 = 𝑤𝜑))
3 excom 1675 . . 3 (∃𝑥𝑤𝑦(𝑦 = 𝑤𝜑) ↔ ∃𝑤𝑥𝑦(𝑦 = 𝑤𝜑))
42, 3bitri 184 . 2 (∃𝑥𝑦𝜑 ↔ ∃𝑤𝑥𝑦(𝑦 = 𝑤𝜑))
5 exsb 2020 . . . 4 (∃𝑥𝑦(𝑦 = 𝑤𝜑) ↔ ∃𝑧𝑥(𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜑)))
6 impexp 263 . . . . . . . 8 (((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑) ↔ (𝑥 = 𝑧 → (𝑦 = 𝑤𝜑)))
76albii 1481 . . . . . . 7 (∀𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑) ↔ ∀𝑦(𝑥 = 𝑧 → (𝑦 = 𝑤𝜑)))
8 19.21v 1884 . . . . . . 7 (∀𝑦(𝑥 = 𝑧 → (𝑦 = 𝑤𝜑)) ↔ (𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜑)))
97, 8bitr2i 185 . . . . . 6 ((𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜑)) ↔ ∀𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
109albii 1481 . . . . 5 (∀𝑥(𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜑)) ↔ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
1110exbii 1616 . . . 4 (∃𝑧𝑥(𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜑)) ↔ ∃𝑧𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
125, 11bitri 184 . . 3 (∃𝑥𝑦(𝑦 = 𝑤𝜑) ↔ ∃𝑧𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
1312exbii 1616 . 2 (∃𝑤𝑥𝑦(𝑦 = 𝑤𝜑) ↔ ∃𝑤𝑧𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
14 excom 1675 . 2 (∃𝑤𝑧𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑) ↔ ∃𝑧𝑤𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
154, 13, 143bitri 206 1 (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1362  wex 1503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546
This theorem depends on definitions:  df-bi 117  df-sb 1774
This theorem is referenced by: (None)
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