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Theorem ax10oe 1656
 Description: Quantifier Substitution for existential quantifiers. Analogue to ax10o 1581 but for ∃ rather than ∀. (Contributed by Jim Kingdon, 21-Dec-2017.)
Assertion
Ref Expression
ax10oe (x x = y → (xψyψ))

Proof of Theorem ax10oe
StepHypRef Expression
1 ax-ia3 101 . . . 4 (x = y → (ψ → (x = y ψ)))
21alimi 1320 . . 3 (x x = yx(ψ → (x = y ψ)))
3 exim 1468 . . 3 (x(ψ → (x = y ψ)) → (xψx(x = y ψ)))
42, 3syl 14 . 2 (x x = y → (xψx(x = y ψ)))
5 ax11e 1655 . . 3 (x = y → (x(x = y ψ) → yψ))
65sps 1408 . 2 (x x = y → (x(x = y ψ) → yψ))
74, 6syld 40 1 (x x = y → (xψyψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1224   = wceq 1226  ∃wex 1358 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 1312  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-11 1374  ax-4 1377  ax-ial 1405 This theorem depends on definitions:  df-bi 110 This theorem is referenced by: (None)
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