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Theorem exsbim 2106
Description: One direction of the equivalence in exsb 2382 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 ax6ev 2077 . . 3 𝑥 𝑥 = 𝑦
2 exim 1932 . . 3 (∀𝑥(𝑥 = 𝑦𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑))
31, 2mpi 20 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)
43exlimiv 2029 1 (∃𝑦𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1654  wex 1878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075
This theorem depends on definitions:  df-bi 199  df-ex 1879
This theorem is referenced by:  eu6im  2646
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