Theorem zfpow 4177

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 4176	. 2 ⊢ ∃𝑥∀𝑦(∀𝑤(𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑧) → 𝑦 ∈ 𝑥)
2		elequ1 2152	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
3		elequ1 2152	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧))
4	2, 3	imbi12d 234	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑧) ↔ (𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧)))
5	4	cbvalv 1917	. . . . 5 ⊢ (∀𝑤(𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑧) ↔ ∀𝑥(𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧))
6	5	imbi1i 238	. . . 4 ⊢ ((∀𝑤(𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑧) → 𝑦 ∈ 𝑥) ↔ (∀𝑥(𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
7	6	albii 1470	. . 3 ⊢ (∀𝑦(∀𝑤(𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑧) → 𝑦 ∈ 𝑥) ↔ ∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
8	7	exbii 1605	. 2 ⊢ (∃𝑥∀𝑦(∀𝑤(𝑤 ∈ 𝑦 → 𝑤 ∈ 𝑧) → 𝑦 ∈ 𝑥) ↔ ∃𝑥∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))
9	1, 8	mpbi 145	1 ⊢ ∃𝑥∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)

Syntax hints:   → wi 4  ∀wal 1351  ∃wex 1492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-13 2150  ax-pow 4176

This theorem depends on definitions:  df-bi 117  df-nf 1461

This theorem is referenced by:  el  4180