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Theorem hbex 1568
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 1425 . . 3 (∃𝑦𝜑 → ∀𝑦𝑦𝜑)
21hbal 1407 . 2 (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝑦𝜑)
3 hbex.1 . . 3 (𝜑 → ∀𝑥𝜑)
4 19.8a 1523 . . 3 (𝜑 → ∃𝑦𝜑)
53, 4alrimih 1399 . 2 (𝜑 → ∀𝑥𝑦𝜑)
62, 5exlimih 1525 1 (∃𝑦𝜑 → ∀𝑥𝑦𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1283  wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  nfex  1569  excomim  1594  19.12  1596  cbvexh  1679  cbvexdh  1843  hbsbv  1859  hbeu1  1952  hbmo  1981  moexexdc  2026
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