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Theorem hbsb2 2347
 Description: Bound-variable hypothesis builder for substitution. (Contributed by NM, 14-May-1993.)
Assertion
Ref Expression
hbsb2 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))

Proof of Theorem hbsb2
StepHypRef Expression
1 sb4 2344 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
2 sb2 2340 . . 3 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
32axc4i 2116 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → ∀𝑥[𝑦 / 𝑥]𝜑)
41, 3syl6 34 1 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473  [wsb 1867 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868 This theorem is referenced by:  nfsb2  2348  hbs1  2424
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