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Theorem speiv 1818
Description: Inference from existential specialization, using implicit substitition. (Contributed by NM, 19-Aug-1993.)
Hypotheses
Ref Expression
speiv.1 (𝑥 = 𝑦 → (𝜑𝜓))
speiv.2 𝜓
Assertion
Ref Expression
speiv 𝑥𝜑
Distinct variable group:   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)

Proof of Theorem speiv
StepHypRef Expression
1 speiv.2 . 2 𝜓
2 speiv.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
32biimprd 157 . . 3 (𝑥 = 𝑦 → (𝜓𝜑))
43spimev 1817 . 2 (𝜓 → ∃𝑥𝜑)
51, 4ax-mp 5 1 𝑥𝜑
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wex 1453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422
This theorem is referenced by: (None)
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