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Theorem ax11e 1776
Description: Analogue to ax-11 1486 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 1775 . . 3 (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
2119.21bi 1538 . 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 103   = wceq 1335  wex 1472
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 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-11 1486  ax-4 1490  ax-ial 1514
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  ax10oe  1777  drex1  1778  sbcof2  1790  ax11ev  1808
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