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Theorem hblem 2880
Description: Change the free variable of a hypothesis builder. Lemma for nfcrii 2906. (Contributed by NM, 21-Jun-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 2591 . 2 ([𝑧 / 𝑦]𝑦𝐴 → ∀𝑥[𝑧 / 𝑦]𝑦𝐴)
3 clelsb3 2878 . 2 ([𝑧 / 𝑦]𝑦𝐴𝑧𝐴)
43albii 1895 . 2 (∀𝑥[𝑧 / 𝑦]𝑦𝐴 ↔ ∀𝑥 𝑧𝐴)
52, 3, 43imtr3i 280 1 (𝑧𝐴 → ∀𝑥 𝑧𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1629  [wsb 2049  wcel 2145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clel 2767
This theorem is referenced by:  nfcrii  2906  bnj1311  31431
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