Users' Mathboxes Mathbox for Wolf Lammen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wl-spae Structured version   Visualization version   GIF version

Theorem wl-spae 33736
Description: Prove an instance of sp 2215 from ax-13 2352 and Tarski's FOL only, without distinct variable conditions. The antecedent 𝑥𝑥 = 𝑦 holds in a multi-object universe only if 𝑦 is substituted for 𝑥, or vice versa, i.e. both variables are effectively the same. The converse ¬ ∀𝑥𝑥 = 𝑦 indicates that both variables are distinct, and it so provides a simple translation of a distinct variable condition to a logical term. In case studies 𝑥𝑥 = 𝑦 and ¬ ∀𝑥𝑥 = 𝑦 can help eliminating distinct variable conditions.

The antecedent 𝑥𝑥 = 𝑦 is expressed in the theorem's name by the abbreviation ae standing for 'all equal'.

Note that we cannot provide a logical predicate telling us directly whether a logical expression contains a particular variable, as such a construct would usually contradict ax-12 2211.

Note that this theorem is also provable from ax-12 2211 alone, so you can pick the axiom it is based on.

Compare this result to 19.3v 2079 and spaev 2145 having distinct variable conditions, but a smaller footprint on axiom usage. (Contributed by Wolf Lammen, 5-Apr-2021.)

Assertion
Ref Expression
wl-spae (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)

Proof of Theorem wl-spae
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 aeveq 2149 . . . . 5 (∀𝑥 𝑥 = 𝑧𝑥 = 𝑦)
21adantl 473 . . . 4 ((𝑦 = 𝑧 ∧ ∀𝑥 𝑥 = 𝑧) → 𝑥 = 𝑦)
32a1d 25 . . 3 ((𝑦 = 𝑧 ∧ ∀𝑥 𝑥 = 𝑧) → (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦))
4 ax13v 2353 . . . . . . 7 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
5 equtrr 2119 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
65al2imi 1910 . . . . . . . 8 (∀𝑥 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑧))
76con3d 149 . . . . . . 7 (∀𝑥 𝑦 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑧 → ¬ ∀𝑥 𝑥 = 𝑦))
84, 7syl6 35 . . . . . 6 𝑥 = 𝑦 → (𝑦 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑧 → ¬ ∀𝑥 𝑥 = 𝑦)))
98com13 88 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑧 → (𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)))
109impcom 396 . . . 4 ((𝑦 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (¬ 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦))
1110con4d 115 . . 3 ((𝑦 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦))
123, 11pm2.61dan 847 . 2 (𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦))
13 ax6evr 2112 . 2 𝑧 𝑦 = 𝑧
1412, 13exlimiiv 2026 1 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wal 1650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-13 2352
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1875
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator