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Theorem ax11e 1789
Description: Analogue to ax-11 1499 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 1788 . . 3 (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
2119.21bi 1551 . 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 1348  wex 1485
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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-11 1499  ax-4 1503  ax-ial 1527
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  ax10oe  1790  drex1  1791  sbcof2  1803  ax11ev  1821
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