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Theorem hblem 2195
Description: Change the free variable of a hypothesis builder. (Contributed by NM, 5-Aug-1993.) (Revised by Andrew Salmon, 11-Jul-2011.)
Hypothesis
Ref Expression
hblem.1 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
Assertion
Ref Expression
hblem (𝑧𝐴 → ∀𝑥 𝑧𝐴)
Distinct variable groups:   𝑦,𝐴   𝑥,𝑧
Allowed substitution hints:   𝐴(𝑥,𝑧)

Proof of Theorem hblem
StepHypRef Expression
1 hblem.1 . . 3 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
21hbsb 1871 . 2 ([𝑧 / 𝑦]𝑦𝐴 → ∀𝑥[𝑧 / 𝑦]𝑦𝐴)
3 clelsb3 2192 . 2 ([𝑧 / 𝑦]𝑦𝐴𝑧𝐴)
43albii 1404 . 2 (∀𝑥[𝑧 / 𝑦]𝑦𝐴 ↔ ∀𝑥 𝑧𝐴)
52, 3, 43imtr3i 198 1 (𝑧𝐴 → ∀𝑥 𝑧𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1287  wcel 1438  [wsb 1692
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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-cleq 2081  df-clel 2084
This theorem is referenced by:  nfcrii  2221
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