Theorem zfpow

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

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

Proof

Step	Hyp	Ref	Expression
1		ax-pow	⊢ ∃ 𝑥 ∀ 𝑦 ( ∀ 𝑤 ( 𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 )
2		elequ1	⊢ ( 𝑤 = 𝑥 → ( 𝑤 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦 ) )
3		elequ1	⊢ ( 𝑤 = 𝑥 → ( 𝑤 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧 ) )
4	2 3	imbi12d	⊢ ( 𝑤 = 𝑥 → ( ( 𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑧 ) ↔ ( 𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧 ) ) )
5	4	cbvalvw	⊢ ( ∀ 𝑤 ( 𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑧 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧 ) )
6	5	imbi1i	⊢ ( ( ∀ 𝑤 ( 𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ↔ ( ∀ 𝑥 ( 𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) )
7	6	albii	⊢ ( ∀ 𝑦 ( ∀ 𝑤 ( 𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ↔ ∀ 𝑦 ( ∀ 𝑥 ( 𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) )
8	7	exbii	⊢ ( ∃ 𝑥 ∀ 𝑦 ( ∀ 𝑤 ( 𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) ↔ ∃ 𝑥 ∀ 𝑦 ( ∀ 𝑥 ( 𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 ) )
9	1 8	mpbi	⊢ ∃ 𝑥 ∀ 𝑦 ( ∀ 𝑥 ( 𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑥 )