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Theorem pm11.57 40712
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 1909 . . . . 5 𝑦𝜑
21nfal 2336 . . . 4 𝑦𝑥𝜑
3 sp 2175 . . . . 5 (∀𝑥𝜑𝜑)
4 stdpc4 2067 . . . . 5 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
53, 4jca 514 . . . 4 (∀𝑥𝜑 → (𝜑 ∧ [𝑦 / 𝑥]𝜑))
62, 5alrimi 2206 . . 3 (∀𝑥𝜑 → ∀𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
76axc4i 2335 . 2 (∀𝑥𝜑 → ∀𝑥𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
8 simpl 485 . . . 4 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝜑)
98sps 2177 . . 3 (∀𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝜑)
109alimi 1806 . 2 (∀𝑥𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑) → ∀𝑥𝜑)
117, 10impbii 211 1 (∀𝑥𝜑 ↔ ∀𝑥𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398  wal 1529  [wsb 2063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-10 2139  ax-11 2154  ax-12 2170
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1775  df-nf 1779  df-sb 2064
This theorem is referenced by: (None)
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