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Theorem exsb 2360
Description: An equivalent expression for existence. One direction (exsbim 1999) needs fewer axioms. (Contributed by NM, 2-Feb-2005.) Avoid ax-13 2375. (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 1912 . 2 𝑦𝜑
2 nfa1 2149 . 2 𝑥𝑥(𝑥 = 𝑦𝜑)
3 ax12v 2176 . . 3 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
4 sp 2181 . . . 4 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
54com12 32 . . 3 (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → 𝜑))
63, 5impbid 212 . 2 (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
71, 2, 6cbvexv1 2343 1 (∃𝑥𝜑 ↔ ∃𝑦𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535  wex 1776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-11 2155  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781
This theorem is referenced by: (None)
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