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Theorem alequexv 2008
Description: Version of equs4v 2007 with its consequence simplified by exsimpr 1871. (Contributed by BJ, 9-Nov-2021.)
Assertion
Ref Expression
alequexv (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem alequexv
StepHypRef Expression
1 ax6ev 1973 . 2 𝑥 𝑥 = 𝑦
2 exim 1835 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑))
31, 2mpi 20 1 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536  wex 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-6 1971
This theorem depends on definitions:  df-bi 210  df-ex 1782
This theorem is referenced by:  exsbim  2009  spsbe  2089  19.8a  2182  bj-spimvwt  34020
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