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Theorem exsb 2347
Description: An equivalent expression for existence. One direction (exsbim 1997) needs fewer axioms. (Contributed by NM, 2-Feb-2005.) Avoid ax-13 2363. (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 1909 . 2 𝑦𝜑
2 nfa1 2140 . 2 𝑥𝑥(𝑥 = 𝑦𝜑)
3 ax12v 2164 . . 3 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
4 sp 2168 . . . 4 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
54com12 32 . . 3 (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → 𝜑))
63, 5impbid 211 . 2 (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
71, 2, 6cbvexv1 2330 1 (∃𝑥𝜑 ↔ ∃𝑦𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1531  wex 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-11 2146  ax-12 2163
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1774  df-nf 1778
This theorem is referenced by: (None)
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