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Theorem bj-alequex 32833
Description: A fol lemma. See bj-alequexv 32780 for a version with a DV condition requiring fewer axioms. Can be used to reduce the proof of spimt 2289 from 133 to 112 bytes. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-alequex (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)

Proof of Theorem bj-alequex
StepHypRef Expression
1 ax6e 2286 . 2 𝑥 𝑥 = 𝑦
2 exim 1801 . 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 1521  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745
This theorem is referenced by:  bj-spimt2  32834  bj-equsal1t  32934
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