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Theorem 2exsb 1901
 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 1900 . . . 4 (∃𝑦𝜑 ↔ ∃𝑤𝑦(𝑦 = 𝑤𝜑))
21exbii 1512 . . 3 (∃𝑥𝑦𝜑 ↔ ∃𝑥𝑤𝑦(𝑦 = 𝑤𝜑))
3 excom 1570 . . 3 (∃𝑥𝑤𝑦(𝑦 = 𝑤𝜑) ↔ ∃𝑤𝑥𝑦(𝑦 = 𝑤𝜑))
42, 3bitri 177 . 2 (∃𝑥𝑦𝜑 ↔ ∃𝑤𝑥𝑦(𝑦 = 𝑤𝜑))
5 exsb 1900 . . . 4 (∃𝑥𝑦(𝑦 = 𝑤𝜑) ↔ ∃𝑧𝑥(𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜑)))
6 impexp 254 . . . . . . . 8 (((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑) ↔ (𝑥 = 𝑧 → (𝑦 = 𝑤𝜑)))
76albii 1375 . . . . . . 7 (∀𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑) ↔ ∀𝑦(𝑥 = 𝑧 → (𝑦 = 𝑤𝜑)))
8 19.21v 1769 . . . . . . 7 (∀𝑦(𝑥 = 𝑧 → (𝑦 = 𝑤𝜑)) ↔ (𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜑)))
97, 8bitr2i 178 . . . . . 6 ((𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜑)) ↔ ∀𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
109albii 1375 . . . . 5 (∀𝑥(𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜑)) ↔ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
1110exbii 1512 . . . 4 (∃𝑧𝑥(𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜑)) ↔ ∃𝑧𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
125, 11bitri 177 . . 3 (∃𝑥𝑦(𝑦 = 𝑤𝜑) ↔ ∃𝑧𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
1312exbii 1512 . 2 (∃𝑤𝑥𝑦(𝑦 = 𝑤𝜑) ↔ ∃𝑤𝑧𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
14 excom 1570 . 2 (∃𝑤𝑧𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑) ↔ ∃𝑧𝑤𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
154, 13, 143bitri 199 1 (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102  ∀wal 1257  ∃wex 1397 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444 This theorem depends on definitions:  df-bi 114  df-sb 1662 This theorem is referenced by: (None)
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