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Theorem hblem 2882
 Description: Change the free variable of a hypothesis builder. (Contributed by NM, 21-Jun-1993.) (Revised by Andrew Salmon, 11-Jul-2011.) Add disjoint variable condition to avoid ax-13 2379. See hblemg 2883 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
Hypothesis
Ref Expression
hblem.1 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
Assertion
Ref Expression
hblem (𝑧𝐴 → ∀𝑥 𝑧𝐴)
Distinct variable groups:   𝑦,𝐴   𝑥,𝑧   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑧)

Proof of Theorem hblem
StepHypRef Expression
1 hblem.1 . . 3 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
21hbsbw 2173 . 2 ([𝑧 / 𝑦]𝑦𝐴 → ∀𝑥[𝑧 / 𝑦]𝑦𝐴)
3 clelsb3 2879 . 2 ([𝑧 / 𝑦]𝑦𝐴𝑧𝐴)
43albii 1821 . 2 (∀𝑥[𝑧 / 𝑦]𝑦𝐴 ↔ ∀𝑥 𝑧𝐴)
52, 3, 43imtr3i 294 1 (𝑧𝐴 → ∀𝑥 𝑧𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536  [wsb 2069   ∈ wcel 2111 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-11 2158 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clel 2830 This theorem is referenced by:  bnj1311  32537
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