Theorem zfpow 5259

Description: Axiom of Power Sets expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)

Assertion

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

Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem zfpow

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-pow 5258	. 2 ⊢ ∃𝑥∀𝑦(∀𝑤(𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑧) → 𝑦 ∈ 𝑥)
2		elequ1 2117	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
3		elequ1 2117	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧))
4	2, 3	imbi12d 347	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑧) ↔ (𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧)))
5	4	cbvalvw 2039	. . . . 5 ⊢ (∀𝑤(𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑧) ↔ ∀𝑥(𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧))
6	5	imbi1i 352	. . . 4 ⊢ ((∀𝑤(𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑧) → 𝑦 ∈ 𝑥) ↔ (∀𝑥(𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
7	6	albii 1816	. . 3 ⊢ (∀𝑦(∀𝑤(𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑧) → 𝑦 ∈ 𝑥) ↔ ∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
8	7	exbii 1844	. 2 ⊢ (∃𝑥∀𝑦(∀𝑤(𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑧) → 𝑦 ∈ 𝑥) ↔ ∃𝑥∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
9	1, 8	mpbi 232	1 ⊢ ∃𝑥∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)

Syntax hints:   → wi 4  ∀wal 1531  ∃wex 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-pow 5258

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777

This theorem is referenced by:  el  5262  axpowndlem2  10014