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Theorem hbex 1509
 Description: If x is not free in φ, it is not free in ∃yφ. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
Hypothesis
Ref Expression
hbex.1 (φxφ)
Assertion
Ref Expression
hbex (yφxyφ)

Proof of Theorem hbex
StepHypRef Expression
1 hbe1 1365 . . 3 (yφyyφ)
21hbal 1346 . 2 (xyφyxyφ)
3 hbex.1 . . 3 (φxφ)
4 19.8a 1464 . . 3 (φyφ)
53, 4alrimih 1338 . 2 (φxyφ)
62, 5exlimih 1466 1 (yφxyφ)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1226  ∃wex 1362 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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  nfex  1510  excomim  1535  19.12  1537  cbvexh  1620  cbvexdh  1783  hbsbv  1799  hbeu1  1892  hbmo  1921  moexexdc  1966
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