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Theorem spimeh 1669
 Description: Existential introduction, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by NM, 3-Feb-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
spimeh.1 (𝜑 → ∀𝑥𝜑)
spimeh.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
spimeh (𝜑 → ∃𝑥𝜓)

Proof of Theorem spimeh
StepHypRef Expression
1 a9e 1627 . 2 𝑥 𝑥 = 𝑦
2 spimeh.1 . . 3 (𝜑 → ∀𝑥𝜑)
3 spimeh.2 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
43com12 30 . . 3 (𝜑 → (𝑥 = 𝑦𝜓))
52, 4eximdh 1543 . 2 (𝜑 → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜓))
61, 5mpi 15 1 (𝜑 → ∃𝑥𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1283   = wceq 1285  ∃wex 1422 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-i9 1464  ax-ial 1468 This theorem depends on definitions:  df-bi 115 This theorem is referenced by: (None)
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