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Theorem equs5 1996
 Description: Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
equs5 x x = y → (x(x = y φ) → x(x = yφ)))

Proof of Theorem equs5
StepHypRef Expression
1 nfnae 1956 . 2 x ¬ x x = y
2 nfa1 1788 . 2 xx(x = yφ)
3 ax11o 1994 . . 3 x x = y → (x = y → (φx(x = yφ))))
43imp3a 420 . 2 x x = y → ((x = y φ) → x(x = yφ)))
51, 2, 4exlimd 1806 1 x x = y → (x(x = y φ) → x(x = 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-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545 This theorem is referenced by:  sb3  2052  sb4  2053
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