 Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-axc10v Structured version   Visualization version   GIF version

Theorem bj-axc10v 33305
 Description: Version of axc10 2349 with a disjoint variable condition, which does not require ax-13 2334. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-axc10v (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-axc10v
StepHypRef Expression
1 ax6v 2022 . . 3 ¬ ∀𝑥 ¬ 𝑥 = 𝑦
2 con3 151 . . . 4 ((𝑥 = 𝑦 → ∀𝑥𝜑) → (¬ ∀𝑥𝜑 → ¬ 𝑥 = 𝑦))
32al2imi 1859 . . 3 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → (∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝑥 = 𝑦))
41, 3mtoi 191 . 2 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → ¬ ∀𝑥 ¬ ∀𝑥𝜑)
5 axc7 2292 . 2 (¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)
64, 5syl 17 1 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1599 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-10 2135  ax-12 2163 This theorem depends on definitions:  df-bi 199  df-ex 1824 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator