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Theorem exsb 1983
 Description: An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.)
Assertion
Ref Expression
exsb (∃𝑥𝜑 ↔ ∃𝑦𝑥(𝑥 = 𝑦𝜑))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem exsb
StepHypRef Expression
1 ax-17 1506 . . 3 (𝜑 → ∀𝑦𝜑)
21sb8eh 1827 . 2 (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
3 sb6 1858 . . 3 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
43exbii 1584 . 2 (∃𝑦[𝑦 / 𝑥]𝜑 ↔ ∃𝑦𝑥(𝑥 = 𝑦𝜑))
52, 4bitri 183 1 (∃𝑥𝜑 ↔ ∃𝑦𝑥(𝑥 = 𝑦𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1329  ∃wex 1468  [wsb 1735 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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514 This theorem depends on definitions:  df-bi 116  df-sb 1736 This theorem is referenced by:  2exsb  1984
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