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Theorem exsb 2337
 Description: An equivalent expression for existence. One direction (exsbim 1989) needs fewer axioms. (Contributed by NM, 2-Feb-2005.) Avoid ax-13 2346. (Revised by Wolf Lammen, 16-Oct-2022.)
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 nfv 1896 . 2 𝑦𝜑
2 nfa1 2123 . 2 𝑥𝑥(𝑥 = 𝑦𝜑)
3 ax12v 2144 . . 3 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
4 sp 2148 . . . 4 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
54com12 32 . . 3 (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → 𝜑))
63, 5impbid 213 . 2 (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
71, 2, 6cbvexv1 2323 1 (∃𝑥𝜑 ↔ ∃𝑦𝑥(𝑥 = 𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1523  ∃wex 1765 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-10 2114  ax-11 2128  ax-12 2143 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ex 1766  df-nf 1770 This theorem is referenced by:  eu6OLD  2621  eu6OLDOLD  2622
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