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Theorem hbex 1527
 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 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
hbex (∃𝑦𝜑 → ∀𝑥𝑦𝜑)

Proof of Theorem hbex
StepHypRef Expression
1 hbe1 1384 . . 3 (∃𝑦𝜑 → ∀𝑦𝑦𝜑)
21hbal 1366 . 2 (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝑦𝜑)
3 hbex.1 . . 3 (𝜑 → ∀𝑥𝜑)
4 19.8a 1482 . . 3 (𝜑 → ∃𝑦𝜑)
53, 4alrimih 1358 . 2 (𝜑 → ∀𝑥𝑦𝜑)
62, 5exlimih 1484 1 (∃𝑦𝜑 → ∀𝑥𝑦𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1241  ∃wex 1381 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 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  nfex  1528  excomim  1553  19.12  1555  cbvexh  1638  cbvexdh  1801  hbsbv  1817  hbeu1  1910  hbmo  1939  moexexdc  1984
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