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Theorem spei 2406
 Description: Inference from existential specialization, using implicit substitution. Remove a distinct variable constraint. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.)
Hypotheses
Ref Expression
spei.1 (𝑥 = 𝑦 → (𝜑𝜓))
spei.2 𝜓
Assertion
Ref Expression
spei 𝑥𝜑

Proof of Theorem spei
StepHypRef Expression
1 ax6e 2395 . 2 𝑥 𝑥 = 𝑦
2 spei.2 . . 3 𝜓
3 spei.1 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3mpbiri 248 . 2 (𝑥 = 𝑦𝜑)
51, 4eximii 1913 1 𝑥𝜑
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∃wex 1853 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-12 2196  ax-13 2391 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854 This theorem is referenced by:  elirrv  8668  bnj1014  31358
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