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Theorem equs5 1707
 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 hbnae 1606 . 2 x x = yx ¬ x x = y)
2 hba1 1430 . 2 (x(x = yφ) → xx(x = yφ))
3 ax11o 1700 . . 3 x x = y → (x = y → (φx(x = yφ))))
43impd 242 . 2 x x = y → ((x = y φ) → x(x = yφ)))
51, 2, 4exlimdh 1484 1 x x = y → (x(x = y φ) → x(x = yφ)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97  ∀wal 1240  ∃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-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643 This theorem is referenced by:  sb3  1709  sb4  1710
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