Theorem axpowg 35454

Description: A generalization of ax-pow 5327 that combines it and zfpow 5328 into a single theorem scheme. Unlike ax-pow 5327, this scheme lacks a distinct variable condition for 𝑦 and 𝑤. (Contributed by BTernaryTau, 26-May-2026.)

Assertion

Ref	Expression
axpowg	⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)

Distinct variable groups:   𝑥,𝑦,𝑧   𝑥,𝑤,𝑧

Proof of Theorem axpowg

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-pow 5327	. 2 ⊢ ∃𝑦∀𝑧(∀𝑣(𝑣 ∈ 𝑧 → 𝑣 ∈ 𝑥) → 𝑧 ∈ 𝑦)
2		elequ1 2152	. . . . . . 7 ⊢ (𝑣 = 𝑤 → (𝑣 ∈ 𝑧 ↔ 𝑤 ∈ 𝑧))
3		elequ1 2152	. . . . . . 7 ⊢ (𝑣 = 𝑤 → (𝑣 ∈ 𝑥 ↔ 𝑤 ∈ 𝑥))
4	2, 3	imbi12d 347	. . . . . 6 ⊢ (𝑣 = 𝑤 → ((𝑣 ∈ 𝑧 → 𝑣 ∈ 𝑥) ↔ (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥)))
5	4	cbvalvw 2059	. . . . 5 ⊢ (∀𝑣(𝑣 ∈ 𝑧 → 𝑣 ∈ 𝑥) ↔ ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥))
6	5	imbi1i 352	. . . 4 ⊢ ((∀𝑣(𝑣 ∈ 𝑧 → 𝑣 ∈ 𝑥) → 𝑧 ∈ 𝑦) ↔ (∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
7	6	albii 1842	. . 3 ⊢ (∀𝑧(∀𝑣(𝑣 ∈ 𝑧 → 𝑣 ∈ 𝑥) → 𝑧 ∈ 𝑦) ↔ ∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
8	7	exbii 1871	. 2 ⊢ (∃𝑦∀𝑧(∀𝑣(𝑣 ∈ 𝑧 → 𝑣 ∈ 𝑥) → 𝑧 ∈ 𝑦) ↔ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
9	1, 8	mpbi 233	1 ⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)

Syntax hints:   → wi 4  ∀wal 1561  ∃wex 1802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-pow 5327

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803

This theorem is referenced by:  axpowg3  35456