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Theorem equs5e 1888
 Description: A property related to substitution that unlike equs5 1996 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
Assertion
Ref Expression
equs5e (x(x = y φ) → x(x = yyφ))

Proof of Theorem equs5e
StepHypRef Expression
1 nfe1 1732 . 2 xx(x = y φ)
2 equs3 1644 . . 3 (x(x = y φ) ↔ ¬ x(x = y → ¬ φ))
3 ax-11 1746 . . . . 5 (x = y → (y ¬ φx(x = y → ¬ φ)))
43con3rr3 128 . . . 4 x(x = y → ¬ φ) → (x = y → ¬ y ¬ φ))
5 df-ex 1542 . . . 4 (yφ ↔ ¬ y ¬ φ)
64, 5syl6ibr 218 . . 3 x(x = y → ¬ φ) → (x = yyφ))
72, 6sylbi 187 . 2 (x(x = y φ) → (x = yyφ))
81, 7alrimi 1765 1 (x(x = y φ) → x(x = yyφ))
 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-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545 This theorem is referenced by:  sb4e  1924
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