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Theorem exsb 1900
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 1435 . . 3 (𝜑 → ∀𝑦𝜑)
21sb8eh 1751 . 2 (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
3 sb6 1782 . . 3 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
43exbii 1512 . 2 (∃𝑦[𝑦 / 𝑥]𝜑 ↔ ∃𝑦𝑥(𝑥 = 𝑦𝜑))
52, 4bitri 177 1 (∃𝑥𝜑 ↔ ∃𝑦𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257  wex 1397  [wsb 1661
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
This theorem depends on definitions:  df-bi 114  df-sb 1662
This theorem is referenced by:  2exsb  1901
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