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Theorem ax11e 1718
 Description: Analogue to ax-11 1438 but for existential quantification. (Contributed by Mario Carneiro and Jim Kingdon, 31-Dec-2017.) (Proved by Mario Carneiro, 9-Feb-2018.)
Assertion
Ref Expression
ax11e (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑦𝜑))

Proof of Theorem ax11e
StepHypRef Expression
1 equs5e 1717 . . 3 (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
2119.21bi 1491 . 2 (∃𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦 → ∃𝑦𝜑))
32com12 30 1 (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑦𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   = 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-11 1438  ax-4 1441  ax-ial 1468 This theorem depends on definitions:  df-bi 115 This theorem is referenced by:  ax10oe  1719  drex1  1720  sbcof2  1732  ax11ev  1750
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