Theorem wl-ax12v2cl 37489

Description: The class version of ax12v2 2180, 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 2179 is a specialization of ax12v2 2180. So any proof using ax12v 2179 will still hold if ax12v2 2180 is used instead.

Theorem ax12v2 2180 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 2178 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 2734	. . 3 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝐴 ↔ 𝑦 = 𝐴))
2	1	cbvexvw 2037	. 2 ⊢ (∃𝑧 𝑧 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)
3		ax12v 2179	. . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
4		eqeq2 2742	. . . . 5 ⊢ (𝑧 = 𝐴 → (𝑥 = 𝑧 ↔ 𝑥 = 𝐴))
5	4	imbi1d 341	. . . . . . 7 ⊢ (𝑧 = 𝐴 → ((𝑥 = 𝑧 → 𝜑) ↔ (𝑥 = 𝐴 → 𝜑)))
6	5	albidv 1920	. . . . . 6 ⊢ (𝑧 = 𝐴 → (∀𝑥(𝑥 = 𝑧 → 𝜑) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))
7	6	imbi2d 340	. . . . 5 ⊢ (𝑧 = 𝐴 → ((𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝐴 → 𝜑))))
8	4, 7	imbi12d 344	. . . 4 ⊢ (𝑧 = 𝐴 → ((𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) ↔ (𝑥 = 𝐴 → (𝜑 → ∀𝑥(𝑥 = 𝐴 → 𝜑)))))
9	3, 8	mpbii 233	. . 3 ⊢ (𝑧 = 𝐴 → (𝑥 = 𝐴 → (𝜑 → ∀𝑥(𝑥 = 𝐴 → 𝜑))))
10	9	exlimiv 1930	. 2 ⊢ (∃𝑧 𝑧 = 𝐴 → (𝑥 = 𝐴 → (𝜑 → ∀𝑥(𝑥 = 𝐴 → 𝜑))))
11	2, 10	sylbir 235	1 ⊢ (∃𝑦 𝑦 = 𝐴 → (𝑥 = 𝐴 → (𝜑 → ∀𝑥(𝑥 = 𝐴 → 𝜑))))

Syntax hints:   → wi 4  ∀wal 1538   = wceq 1540  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2722

This theorem is referenced by: (None)