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Theorem axi11e 2332
 Description: Axiom of Variable Substitution for Existence (intuitionistic logic axiom ax-i11e). This can be derived from ax-11 1746 in a classical context but a separate axiom is needed for intuitionistic predicate calculus. (Contributed by Jim Kingdon, 31-Dec-2017.)
Assertion
Ref Expression
axi11e (x = y → (x(x = y φ) → yφ))

Proof of Theorem axi11e
StepHypRef Expression
1 ax-11 1746 . . 3 (x = y → (y ¬ φx(x = y → ¬ φ)))
2 alnex 1543 . . 3 (y ¬ φ ↔ ¬ yφ)
3 alinexa 1578 . . 3 (x(x = y → ¬ φ) ↔ ¬ x(x = y φ))
41, 2, 33imtr3g 260 . 2 (x = y → (¬ yφ → ¬ x(x = y φ)))
54con4d 97 1 (x = y → (x(x = y φ) → yφ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542 This theorem is referenced by: (None)
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