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Theorem pm11.57 39109
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 1992 . . . . 5 𝑦𝜑
21nfal 2300 . . . 4 𝑦𝑥𝜑
3 sp 2200 . . . . 5 (∀𝑥𝜑𝜑)
4 stdpc4 2490 . . . . 5 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
53, 4jca 555 . . . 4 (∀𝑥𝜑 → (𝜑 ∧ [𝑦 / 𝑥]𝜑))
62, 5alrimi 2229 . . 3 (∀𝑥𝜑 → ∀𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
76axc4i 2278 . 2 (∀𝑥𝜑 → ∀𝑥𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
8 simpl 474 . . . 4 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝜑)
98sps 2202 . . 3 (∀𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝜑)
109alimi 1888 . 2 (∀𝑥𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑) → ∀𝑥𝜑)
117, 10impbii 199 1 (∀𝑥𝜑 ↔ ∀𝑥𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wal 1630  [wsb 2046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1854  df-nf 1859  df-sb 2047
This theorem is referenced by: (None)
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