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Theorem equs5e 1673
Description: A property related to substitution that unlike equs5 1707 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) (Revised by NM, 3-Feb-2015.)
Assertion
Ref Expression
equs5e (x(x = y φ) → x(x = yyφ))

Proof of Theorem equs5e
StepHypRef Expression
1 19.8a 1479 . . . . 5 (φyφ)
2 hbe1 1381 . . . . 5 (yφyyφ)
31, 2syl 14 . . . 4 (φyyφ)
43anim2i 324 . . 3 ((x = y φ) → (x = y yyφ))
54eximi 1488 . 2 (x(x = y φ) → x(x = y yyφ))
6 equs5a 1672 . 2 (x(x = y yyφ) → x(x = yyφ))
75, 6syl 14 1 (x(x = y φ) → x(x = yyφ))
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-11 1394  ax-4 1397  ax-ial 1424
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  ax11e  1674  sb4e  1683
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