 Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-axc11v Structured version   Visualization version   GIF version

Theorem bj-axc11v 32872
 Description: Version of axc11 2347 with a dv condition, which does not require ax-13 2282 nor ax-10 2059. Remark: the following theorems (hbae 2348, nfae 2349, hbnae 2350, nfnae 2351, hbnaes 2352) would need to be totally unbundled to be proved without ax-13 2282, hence would be simple consequences of ax-5 1879 or nfv 1883. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-axc11v (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-axc11v
StepHypRef Expression
1 axc11r 2223 . 2 (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
21bj-aecomsv 32871 1 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1521 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745 This theorem is referenced by:  bj-dral1v  32873
 Copyright terms: Public domain W3C validator