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Theorem pm11.59 38112
 Description: Theorem *11.59 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)
Assertion
Ref Expression
pm11.59 (∀𝑥(𝜑𝜓) → ∀𝑦𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜓 ∧ [𝑦 / 𝑥]𝜓)))
Distinct variable groups:   𝜑,𝑦   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem pm11.59
StepHypRef Expression
1 nfv 1840 . . 3 𝑦(𝜑𝜓)
21nfal 2150 . 2 𝑦𝑥(𝜑𝜓)
3 sp 2051 . . . 4 (∀𝑥(𝜑𝜓) → (𝜑𝜓))
4 spsbim 2393 . . . 4 (∀𝑥(𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
53, 4anim12d 585 . . 3 (∀𝑥(𝜑𝜓) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜓 ∧ [𝑦 / 𝑥]𝜓)))
65axc4i 2127 . 2 (∀𝑥(𝜑𝜓) → ∀𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜓 ∧ [𝑦 / 𝑥]𝜓)))
72, 6alrimi 2080 1 (∀𝑥(𝜑𝜓) → ∀𝑦𝑥((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜓 ∧ [𝑦 / 𝑥]𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1478  [wsb 1877 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707  df-sb 1878 This theorem is referenced by: (None)
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