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Theorem exsb 2130
 Description: An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.)
Assertion
Ref Expression
exsb (xφyx(x = yφ))
Distinct variable groups:   x,y   φ,y
Allowed substitution hint:   φ(x)

Proof of Theorem exsb
StepHypRef Expression
1 nfv 1619 . 2 yφ
2 nfa1 1788 . 2 xx(x = yφ)
3 ax11v 2096 . . 3 (x = y → (φx(x = yφ)))
4 sp 1747 . . . 4 (x(x = yφ) → (x = yφ))
54com12 27 . . 3 (x = y → (x(x = yφ) → φ))
63, 5impbid 183 . 2 (x = y → (φx(x = yφ)))
71, 2, 6cbvex 1985 1 (xφyx(x = yφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545 This theorem is referenced by:  2exsb  2132
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