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Theorem wl-spae 32965
Description: Prove an instance of sp 2051 from ax-13 2245 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 2044.

Note that this theorem is also provable from ax-12 2044 alone, so you can pick the axiom it is based on.

Compare this result to 19.3v 1894 and spaev 1975 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 1979 . . . . 5 (∀𝑥 𝑥 = 𝑧𝑥 = 𝑦)
21adantl 482 . . . 4 ((𝑦 = 𝑧 ∧ ∀𝑥 𝑥 = 𝑧) → 𝑥 = 𝑦)
32a1d 25 . . 3 ((𝑦 = 𝑧 ∧ ∀𝑥 𝑥 = 𝑧) → (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦))
4 ax13v 2246 . . . . . . 7 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
5 equtrr 1946 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
65al2imi 1740 . . . . . . . 8 (∀𝑥 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑧))
76con3d 148 . . . . . . 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 446 . . . 4 ((𝑦 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (¬ 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦))
1110con4d 114 . . 3 ((𝑦 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦))
123, 11pm2.61dan 831 . 2 (𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦))
13 ax6evr 1939 . 2 𝑧 𝑦 = 𝑧
1412, 13exlimiiv 1856 1 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wal 1478
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-13 2245
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702
This theorem is referenced by: (None)
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