Theorem wl-ax12v2cl 37868

Description: The class version of ax12v2 2191, where the set variable 𝑦 is replaced with the class variable 𝐴. This is possible if 𝐴 is known to be a set, expressed by the antecedent.

Theorem ax12v 2190 is a specialization of ax12v2 2191. So any proof using ax12v 2190 will still hold if ax12v2 2191 is used instead.

Theorem ax12v2 2191 expresses that two equal set variables cannot be distinguished by whatever complicated formula 𝜑 if one is replaced with the other in it. This theorem states a similar result for a class variable known to be a set: All sets equal to the class variable behave the same if they replace the class variable in 𝜑.

Most axioms in FOL containing an equation correspond to a theorem where a class variable known to be a set replaces a set variable in the formula. Some exceptions cannot be avoided: The set variable must nowhere be bound. And it is not possible to state a distinct variable condition where a class 𝐴 is different from another, or distinct from a variable with type wff. So ax-12 2189 proper is out of reach: you cannot replace 𝑦 in ∀𝑦𝜑 with a class variable.

But where such limitations are not violated, the proof of the FOL theorem should carry over to a version where a class variable, known to be set, appears instead of a set variable. (Contributed by Wolf Lammen, 8-Aug-2020.)

Assertion

Ref	Expression
wl-ax12v2cl	⊢ (∃𝑦 𝑦 = 𝐴 → (𝑥 = 𝐴 → (𝜑 → ∀𝑥(𝑥 = 𝐴 → 𝜑))))

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem wl-ax12v2cl

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2743	. . 3 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝐴 ↔ 𝑦 = 𝐴))
2	1	cbvexvw 2044	. 2 ⊢ (∃𝑧 𝑧 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)
3		ax12v 2190	. . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
4		eqeq2 2751	. . . . 5 ⊢ (𝑧 = 𝐴 → (𝑥 = 𝑧 ↔ 𝑥 = 𝐴))
5	4	imbi1d 342	. . . . . . 7 ⊢ (𝑧 = 𝐴 → ((𝑥 = 𝑧 → 𝜑) ↔ (𝑥 = 𝐴 → 𝜑)))
6	5	albidv 1927	. . . . . 6 ⊢ (𝑧 = 𝐴 → (∀𝑥(𝑥 = 𝑧 → 𝜑) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))
7	6	imbi2d 341	. . . . 5 ⊢ (𝑧 = 𝐴 → ((𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝐴 → 𝜑))))
8	4, 7	imbi12d 345	. . . 4 ⊢ (𝑧 = 𝐴 → ((𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) ↔ (𝑥 = 𝐴 → (𝜑 → ∀𝑥(𝑥 = 𝐴 → 𝜑)))))
9	3, 8	mpbii 234	. . 3 ⊢ (𝑧 = 𝐴 → (𝑥 = 𝐴 → (𝜑 → ∀𝑥(𝑥 = 𝐴 → 𝜑))))
10	9	exlimiv 1937	. 2 ⊢ (∃𝑧 𝑧 = 𝐴 → (𝑥 = 𝐴 → (𝜑 → ∀𝑥(𝑥 = 𝐴 → 𝜑))))
11	2, 10	sylbir 236	1 ⊢ (∃𝑦 𝑦 = 𝐴 → (𝑥 = 𝐴 → (𝜑 → ∀𝑥(𝑥 = 𝐴 → 𝜑))))

Syntax hints:   → wi 4  ∀wal 1545   = wceq 1547  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-9 2129  ax-12 2189  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-cleq 2731

This theorem is referenced by: (None)