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Theorem alequexv 2007
Description: Version of equs4v 2006 with its consequence simplified by exsimpr 1875. (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 1976 . 2 𝑥 𝑥 = 𝑦
2 exim 1839 . 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 1539  wex 1785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-6 1974
This theorem depends on definitions:  df-bi 206  df-ex 1786
This theorem is referenced by:  exsbim  2008  spsbe  2088  19.8a  2177  bj-spimvwt  34829
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