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Theorem exsbim 1999
Description: One direction of the equivalence in exsb 2360 is based on fewer axioms. (Contributed by Wolf Lammen, 2-Mar-2023.)
Assertion
Ref Expression
exsbim (∃𝑦𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem exsbim
StepHypRef Expression
1 alequexv 1998 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)
21exlimiv 1928 1 (∃𝑦𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  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
This theorem depends on definitions:  df-bi 207  df-ex 1777
This theorem is referenced by:  spsbe  2080  eu6  2572  eu6im  2573  eu6w  42663
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