Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  equs4 Structured version   GIF version

Theorem equs4 1610
 Description: Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.)
Assertion
Ref Expression
equs4 (x(x = yφ) → x(x = y φ))

Proof of Theorem equs4
StepHypRef Expression
1 a9e 1583 . . 3 x x = y
2 19.29 1508 . . 3 ((x(x = yφ) x x = y) → x((x = yφ) x = y))
31, 2mpan2 401 . 2 (x(x = yφ) → x((x = yφ) x = y))
4 ancl 301 . . . 4 ((x = yφ) → (x = y → (x = y φ)))
54imp 115 . . 3 (((x = yφ) x = y) → (x = y φ))
65eximi 1488 . 2 (x((x = yφ) x = y) → x(x = y φ))
73, 6syl 14 1 (x(x = yφ) → x(x = y φ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1240   = wceq 1242  ∃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-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-i9 1420  ax-ial 1424 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  sb2  1647  equs45f  1680
 Copyright terms: Public domain W3C validator