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Theorem hbex 1598
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 1454 . . 3 (∃𝑦𝜑 → ∀𝑦𝑦𝜑)
21hbal 1436 . 2 (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝑦𝜑)
3 hbex.1 . . 3 (𝜑 → ∀𝑥𝜑)
4 19.8a 1552 . . 3 (𝜑 → ∃𝑦𝜑)
53, 4alrimih 1428 . 2 (𝜑 → ∀𝑥𝑦𝜑)
62, 5exlimih 1555 1 (∃𝑦𝜑 → ∀𝑥𝑦𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1312  wex 1451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  nfex  1599  excomim  1624  19.12  1626  cbvexh  1711  cbvexdh  1876  hbsbv  1892  hbeu1  1985  hbmo  2014  moexexdc  2059
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