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Theorem zfpow 4993
Description: Axiom of Power Sets expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
Assertion
Ref Expression
zfpow 𝑥𝑦(∀𝑥(𝑥𝑦𝑥𝑧) → 𝑦𝑥)
Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem zfpow
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ax-pow 4992 . 2 𝑥𝑦(∀𝑤(𝑤𝑦𝑤𝑧) → 𝑦𝑥)
2 elequ1 2146 . . . . . . 7 (𝑤 = 𝑥 → (𝑤𝑦𝑥𝑦))
3 elequ1 2146 . . . . . . 7 (𝑤 = 𝑥 → (𝑤𝑧𝑥𝑧))
42, 3imbi12d 333 . . . . . 6 (𝑤 = 𝑥 → ((𝑤𝑦𝑤𝑧) ↔ (𝑥𝑦𝑥𝑧)))
54cbvalv 2418 . . . . 5 (∀𝑤(𝑤𝑦𝑤𝑧) ↔ ∀𝑥(𝑥𝑦𝑥𝑧))
65imbi1i 338 . . . 4 ((∀𝑤(𝑤𝑦𝑤𝑧) → 𝑦𝑥) ↔ (∀𝑥(𝑥𝑦𝑥𝑧) → 𝑦𝑥))
76albii 1896 . . 3 (∀𝑦(∀𝑤(𝑤𝑦𝑤𝑧) → 𝑦𝑥) ↔ ∀𝑦(∀𝑥(𝑥𝑦𝑥𝑧) → 𝑦𝑥))
87exbii 1923 . 2 (∃𝑥𝑦(∀𝑤(𝑤𝑦𝑤𝑧) → 𝑦𝑥) ↔ ∃𝑥𝑦(∀𝑥(𝑥𝑦𝑥𝑧) → 𝑦𝑥))
91, 8mpbi 220 1 𝑥𝑦(∀𝑥(𝑥𝑦𝑥𝑧) → 𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1630  wex 1853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-11 2183  ax-12 2196  ax-13 2391  ax-pow 4992
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854
This theorem is referenced by:  el  4996  axpowndlem2  9612
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