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Theorem sbhb 2541
 Description: Two ways of expressing "𝑥 is (effectively) not free in 𝜑". Usage of this theorem is discouraged because it depends on ax-13 2380. (Contributed by NM, 29-May-2009.) (New usage is discouraged.)
Assertion
Ref Expression
sbhb ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑))
Distinct variable group:   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem sbhb
StepHypRef Expression
1 nfv 1916 . . . 4 𝑦𝜑
21sb8 2537 . . 3 (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
32imbi2i 340 . 2 ((𝜑 → ∀𝑥𝜑) ↔ (𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑))
4 19.21v 1941 . 2 (∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑) ↔ (𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑))
53, 4bitr4i 281 1 ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1537  [wsb 2070 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2143  ax-11 2159  ax-12 2176  ax-13 2380 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-ex 1783  df-nf 1787  df-sb 2071 This theorem is referenced by: (None)
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