Theorem wl-ax12v2cl 37761

Description: The class version of ax12v2 2187, 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 2186 is a specialization of ax12v2 2187. So any proof using ax12v 2186 will still hold if ax12v2 2187 is used instead.

Theorem ax12v2 2187 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 2185 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 2741	. . 3 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝐴 ↔ 𝑦 = 𝐴))
2	1	cbvexvw 2039	. 2 ⊢ (∃𝑧 𝑧 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)
3		ax12v 2186	. . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
4		eqeq2 2749	. . . . 5 ⊢ (𝑧 = 𝐴 → (𝑥 = 𝑧 ↔ 𝑥 = 𝐴))
5	4	imbi1d 341	. . . . . . 7 ⊢ (𝑧 = 𝐴 → ((𝑥 = 𝑧 → 𝜑) ↔ (𝑥 = 𝐴 → 𝜑)))
6	5	albidv 1922	. . . . . 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 1932	. 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 1540   = wceq 1542  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2729

This theorem is referenced by: (None)