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Theorem ax9o 1585
 Description: An implication related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
Assertion
Ref Expression
ax9o (x(x = yxφ) → φ)

Proof of Theorem ax9o
StepHypRef Expression
1 a9e 1583 . 2 x x = y
2 19.29r 1509 . . 3 ((x x = y x(x = yxφ)) → x(x = y (x = yxφ)))
3 hba1 1430 . . . . 5 (xφxxφ)
4 pm3.35 329 . . . . 5 ((x = y (x = yxφ)) → xφ)
53, 4exlimih 1481 . . . 4 (x(x = y (x = yxφ)) → xφ)
6 ax-4 1397 . . . 4 (xφφ)
75, 6syl 14 . . 3 (x(x = y (x = yxφ)) → φ)
82, 7syl 14 . 2 ((x x = y x(x = yxφ)) → φ)
91, 8mpan 400 1 (x(x = yxφ) → 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:  equsalh  1611  spimth  1620  spimh  1622
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