Theorem wl-spae 34763

Description: Prove an instance of sp 2182 from ax-13 2390 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 2177.

Note that this theorem is also provable from ax-12 2177 alone, so you can pick the axiom it is based on.

Compare this result to 19.3v 1986 and spaev 2057 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 = 𝑦 → 𝑥 = 𝑦)

Proof of Theorem wl-spae

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		aeveq 2061	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑧 → 𝑥 = 𝑦)
2	1	adantl 484	. . . 4 ⊢ ((𝑦 = 𝑧 ∧ ∀𝑥 𝑥 = 𝑧) → 𝑥 = 𝑦)
3	2	a1d 25	. . 3 ⊢ ((𝑦 = 𝑧 ∧ ∀𝑥 𝑥 = 𝑧) → (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦))
4		ax13v 2391	. . . . . . 7 ⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
5		equtrr 2029	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑦 → 𝑥 = 𝑧))
6	5	al2imi 1816	. . . . . . . 8 ⊢ (∀𝑥 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑧))
7	6	con3d 155	. . . . . . 7 ⊢ (∀𝑥 𝑦 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑧 → ¬ ∀𝑥 𝑥 = 𝑦))
8	4, 7	syl6 35	. . . . . 6 ⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → (¬ ∀𝑥 𝑥 = 𝑧 → ¬ ∀𝑥 𝑥 = 𝑦)))
9	8	com13 88	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → (𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)))
10	9	impcom 410	. . . 4 ⊢ ((𝑦 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (¬ 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦))
11	10	con4d 115	. . 3 ⊢ ((𝑦 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦))
12	3, 11	pm2.61dan 811	. 2 ⊢ (𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦))
13		ax6evr 2022	. 2 ⊢ ∃𝑧 𝑦 = 𝑧
14	12, 13	exlimiiv 1932	1 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781

This theorem is referenced by: (None)