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Theorem speiv 1742
 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 147 . . 3 (𝑥 = 𝑦 → (𝜓𝜑))
43spimev 1741 . 2 (𝜓 → ∃𝑥𝜑)
51, 4ax-mp 7 1 𝑥𝜑
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∃wex 1381 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350 This theorem is referenced by: (None)
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