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Theorem hblem 2613
Description: Change the free variable of a hypothesis builder. Lemma for nfcrii 2639. (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 2324 . 2 ([𝑧 / 𝑦]𝑦𝐴 → ∀𝑥[𝑧 / 𝑦]𝑦𝐴)
3 clelsb3 2611 . 2 ([𝑧 / 𝑦]𝑦𝐴𝑧𝐴)
43albii 1720 . 2 (∀𝑥[𝑧 / 𝑦]𝑦𝐴 ↔ ∀𝑥 𝑧𝐴)
52, 3, 43imtr3i 275 1 (𝑧𝐴 → ∀𝑥 𝑧𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1466  [wsb 1828  wcel 1937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485
This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-ex 1693  df-nf 1697  df-sb 1829  df-cleq 2498  df-clel 2501
This theorem is referenced by:  nfcrii  2639  bnj1311  29985
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