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Theorem ax11e 1674
 Description: Analogue to ax-11 1394 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 = y φ) → yφ))

Proof of Theorem ax11e
StepHypRef Expression
1 equs5e 1673 . . 3 (x(x = y φ) → x(x = yyφ))
2119.21bi 1447 . 2 (x(x = y φ) → (x = yyφ))
32com12 27 1 (x = y → (x(x = y φ) → yφ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242  ∃wex 1378 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 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-11 1394  ax-4 1397  ax-ial 1424 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  ax10oe  1675  drex1  1676  sbcof2  1688  ax11ev  1706
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