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Metamath Proof Explorer

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 )