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Theorem pm11.57 37611
 Description: Theorem *11.57 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
pm11.57 (∀𝑥𝜑 ↔ ∀𝑥𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
Distinct variable group:   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem pm11.57
StepHypRef Expression
1 nfv 1830 . . . . 5 𝑦𝜑
21nfal 2139 . . . 4 𝑦𝑥𝜑
3 sp 2041 . . . . 5 (∀𝑥𝜑𝜑)
4 stdpc4 2341 . . . . 5 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
53, 4jca 553 . . . 4 (∀𝑥𝜑 → (𝜑 ∧ [𝑦 / 𝑥]𝜑))
62, 5alrimi 2069 . . 3 (∀𝑥𝜑 → ∀𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
76axc4i 2116 . 2 (∀𝑥𝜑 → ∀𝑥𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
8 simpl 472 . . . 4 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝜑)
98sps 2043 . . 3 (∀𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝜑)
109alimi 1730 . 2 (∀𝑥𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑) → ∀𝑥𝜑)
117, 10impbii 198 1 (∀𝑥𝜑 ↔ ∀𝑥𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383  ∀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-11 2021  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696  df-nf 1701  df-sb 1868 This theorem is referenced by: (None)
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